\chapter{Doubly indexed recurrences}

%\section{Splitting $\mathcal{C}$, the Catalan triangle}

A Riordan array $\mathcal{R}$ is usually denoted by a pair
of functions $(d(t), h(t))$, with the main request $h(0)=0$;
moreover, an array $\mathcal{R}$ is \emph{proper} if $d(0)\neq 0$
and $\left.\frac{\partial h(t)}{\partial t}\right|_{t=0}\neq0$.
It is possible to characterize coefficients lying on column $k$
-- indexing is $0$-based, conventionally -- with the following
gf:
\begin{displaymath}
    d(t)h(t)^{k}
\end{displaymath}

Assume to start with a purely generic matrix:
\begin{equation}
\left[\begin{matrix}
c_{0,0} &  &  &  &  &  &  &  &  &  \\
c_{1,0} & c_{1,1} &  &  &  &  &  &  &  &  \\
c_{2,0} & c_{2,1} & c_{2,2} &  &  &  &  &  &  &  \\
c_{3,0} & c_{3,1} & c_{3,2} & c_{3,3} &  &  &  &  &  &  \\
c_{4,0} & c_{4,1} & c_{4,2} & c_{4,3} & c_{4,4} &  &  &  &  &  \\
c_{5,0} & c_{5,1} & c_{5,2} & c_{5,3} & c_{5,4} & c_{5,5} &  &  &  &  \\
c_{6,0} & c_{6,1} & c_{6,2} & c_{6,3} & c_{6,4} & c_{6,5} & c_{6,6} &  &  &  \\
c_{7,0} & c_{7,1} & c_{7,2} & c_{7,3} & c_{7,4} & c_{7,5} & c_{7,6} & c_{7,7} &  &  \\
c_{8,0} & c_{8,1} & c_{8,2} & c_{8,3} & c_{8,4} & c_{8,5} & c_{8,6} & c_{8,7} & c_{8,8} &  \\
c_{9,0} & c_{9,1} & c_{9,2} & c_{9,3} & c_{9,4} & c_{9,5} & c_{9,6} & c_{9,7} & c_{9,8} & c_{9,9} \\
\end{matrix}\right]
\label{eq:purely:generic:catalan}
\end{equation}
and unfold each term starting from row $1$ according to the Catalan recurrence:
\begin{equation}
c_{n + 1,k + 1} = c_{n,k} + c_{n,k + 1} + c_{n,k + 2} + c_{n,k + 3} + c_{n,k + 4} + c_{n,k + 5} + c_{n,k + 6} + c_{n,k + 7} + c_{n,k + 8} + c_{n,k + 9}
\label{eq:catalan:rec}
\end{equation}
the result is the following matrix, depending only on $c_{0,0}$, \textcolor{blue}{blue} colored:
\begin{equation}
\left[\begin{matrix}
\textcolor{blue}{c_{0,0}} &  &  &  &  &  &  &  &  &  \\
c_{0,0} & c_{0,0} &  &  &  &  &  &  &  &  \\
2 c_{0,0} & 2 c_{0,0} & c_{0,0} &  &  &  &  &  &  &  \\
5 c_{0,0} & 5 c_{0,0} & 3 c_{0,0} & c_{0,0} &  &  &  &  &  &  \\
14 c_{0,0} & 14 c_{0,0} & 9 c_{0,0} & 4 c_{0,0} & c_{0,0} &  &  &  &  &  \\
42 c_{0,0} & 42 c_{0,0} & 28 c_{0,0} & 14 c_{0,0} & 5 c_{0,0} & c_{0,0} &  &  &  &  \\
132 c_{0,0} & 132 c_{0,0} & 90 c_{0,0} & 48 c_{0,0} & 20 c_{0,0} & 6 c_{0,0} & c_{0,0} &  &  &  \\
429 c_{0,0} & 429 c_{0,0} & 297 c_{0,0} & 165 c_{0,0} & 75 c_{0,0} & 27 c_{0,0} & 7 c_{0,0} & c_{0,0} &  &  \\
1430 c_{0,0} & 1430 c_{0,0} & 1001 c_{0,0} & 572 c_{0,0} & 275 c_{0,0} & 110 c_{0,0} & 35 c_{0,0} & 8 c_{0,0} & c_{0,0} &  \\
4862 c_{0,0} & 4862 c_{0,0} & 3432 c_{0,0} & 2002 c_{0,0} & 1001 c_{0,0} & 429 c_{0,0} & 154 c_{0,0} & 44 c_{0,0} & 9 c_{0,0} & c_{0,0} \\
\end{matrix}\right]
\label{eq:unfolded:catalan}
\end{equation}
to obtain a standard triangle of numbers, known as Catalan triangle, just plug $c_{0,0}=1$
in the above matrix:
\begin{equation}
\left[\begin{matrix}
1 &  &  &  &  &  &  &  &  &  \\
1 & 1 &  &  &  &  &  &  &  &  \\
2 & 2 & 1 &  &  &  &  &  &  &  \\
5 & 5 & 3 & 1 &  &  &  &  &  &  \\
14 & 14 & 9 & 4 & 1 &  &  &  &  &  \\
42 & 42 & 28 & 14 & 5 & 1 &  &  &  &  \\
132 & 132 & 90 & 48 & 20 & 6 & 1 &  &  &  \\
429 & 429 & 297 & 165 & 75 & 27 & 7 & 1 &  &  \\
1430 & 1430 & 1001 & 572 & 275 & 110 & 35 & 8 & 1 &  \\
4862 & 4862 & 3432 & 2002 & 1001 & 429 & 154 & 44 & 9 & 1 \\
\end{matrix}\right]
\label{eq:standard:catalan}
\end{equation}
Such triangle is the matrix expansion of the Riordan array $\mathcal{C}$ defined as:
\begin{displaymath}
    \mathcal{C}=\left(\frac{1-\sqrt{1-4\,t}}{2\,t},
        \frac{1-\sqrt{1-4\,t}}{2}\right)
\end{displaymath}

\subsection{Two-way splitting}

Now let us unfold starting from row $2$, leaving the first \emph{two} rows purely generic,
producing matrix in \autoref{eq:two:splitted:catalan}. We see that the lower part of the
matrix -- coefficients lying under row $2$, such row included -- dependends on $\textcolor{blue}{c_{1,0}}$
and $\textcolor{red}{c_{1,1}}$: but it should be possible to rewrite $\textcolor{red}{c_{1,1}}$
according \autoref{eq:catalan:rec}, yielding matrix in \autoref{eq:clean:two:splitted:catalan},
which depends only on $\textcolor{blue}{c_{0,0}}$ and $\textcolor{blue}{c_{1,0}}$.
\begin{sidewaystable}
%\vskip-4cm
\scriptsize
%\rotatebox{90}{$
\begin{align}
\left[\begin{matrix}
\textcolor{blue}{c_{0,0}} &  &  &  &  &  &  &  &  &  \\
\textcolor{blue}{c_{1,0}} & \textcolor{red}{c_{1,1}} &  &  &  &  &  &  &  &  \\
c_{1,0} + c_{1,1} & c_{1,0} + c_{1,1} & c_{1,1} &  &  &  &  &  &  &  \\
2 c_{1,0} + 3 c_{1,1} & 2 c_{1,0} + 3 c_{1,1} & c_{1,0} + 2 c_{1,1} & c_{1,1} &  &  &  &  &  &  \\
5 c_{1,0} + 9 c_{1,1} & 5 c_{1,0} + 9 c_{1,1} & 3 c_{1,0} + 6 c_{1,1} & c_{1,0} + 3 c_{1,1} & c_{1,1} &  &  &  &  &  \\
14 c_{1,0} + 28 c_{1,1} & 14 c_{1,0} + 28 c_{1,1} & 9 c_{1,0} + 19 c_{1,1} & 4 c_{1,0} + 10 c_{1,1} & c_{1,0} + 4 c_{1,1} & c_{1,1} &  &  &  &  \\
42 c_{1,0} + 90 c_{1,1} & 42 c_{1,0} + 90 c_{1,1} & 28 c_{1,0} + 62 c_{1,1} & 14 c_{1,0} + 34 c_{1,1} & 5 c_{1,0} + 15 c_{1,1} & c_{1,0} + 5 c_{1,1} & c_{1,1} &  &  &  \\
132 c_{1,0} + 297 c_{1,1} & 132 c_{1,0} + 297 c_{1,1} & 90 c_{1,0} + 207 c_{1,1} & 48 c_{1,0} + 117 c_{1,1} & 20 c_{1,0} + 55 c_{1,1} & 6 c_{1,0} + 21 c_{1,1} & c_{1,0} + 6 c_{1,1} & c_{1,1} &  &  \\
429 c_{1,0} + 1001 c_{1,1} & 429 c_{1,0} + 1001 c_{1,1} & 297 c_{1,0} + 704 c_{1,1} & 165 c_{1,0} + 407 c_{1,1} & 75 c_{1,0} + 200 c_{1,1} & 27 c_{1,0} + 83 c_{1,1} & 7 c_{1,0} + 28 c_{1,1} & c_{1,0} + 7 c_{1,1} & c_{1,1} &  \\
1430 c_{1,0} + 3432 c_{1,1} & 1430 c_{1,0} + 3432 c_{1,1} & 1001 c_{1,0} + 2431 c_{1,1} & 572 c_{1,0} + 1430 c_{1,1} & 275 c_{1,0} + 726 c_{1,1} & 110 c_{1,0} + 319 c_{1,1} & 35 c_{1,0} + 119 c_{1,1} & 8 c_{1,0} + 36 c_{1,1} & c_{1,0} + 8 c_{1,1} & c_{1,1} \\
\end{matrix}\right]
\label{eq:two:splitted:catalan} \\
\left[\begin{matrix}
\textcolor{blue}{c_{0,0}} &  &  &  &  &  &  &  &  &  \\
\textcolor{blue}{c_{1,0}} & c_{0,0} &  &  &  &  &  &  &  &  \\
c_{0,0} + c_{1,0} & c_{0,0} + c_{1,0} & c_{0,0} &  &  &  &  &  &  &  \\
3 c_{0,0} + 2 c_{1,0} & 3 c_{0,0} + 2 c_{1,0} & 2 c_{0,0} + c_{1,0} & c_{0,0} &  &  &  &  &  &  \\
9 c_{0,0} + 5 c_{1,0} & 9 c_{0,0} + 5 c_{1,0} & 6 c_{0,0} + 3 c_{1,0} & 3 c_{0,0} + c_{1,0} & c_{0,0} &  &  &  &  &  \\
28 c_{0,0} + 14 c_{1,0} & 28 c_{0,0} + 14 c_{1,0} & 19 c_{0,0} + 9 c_{1,0} & 10 c_{0,0} + 4 c_{1,0} & 4 c_{0,0} + c_{1,0} & c_{0,0} &  &  &  &  \\
90 c_{0,0} + 42 c_{1,0} & 90 c_{0,0} + 42 c_{1,0} & 62 c_{0,0} + 28 c_{1,0} & 34 c_{0,0} + 14 c_{1,0} & 15 c_{0,0} + 5 c_{1,0} & 5 c_{0,0} + c_{1,0} & c_{0,0} &  &  &  \\
297 c_{0,0} + 132 c_{1,0} & 297 c_{0,0} + 132 c_{1,0} & 207 c_{0,0} + 90 c_{1,0} & 117 c_{0,0} + 48 c_{1,0} & 55 c_{0,0} + 20 c_{1,0} & 21 c_{0,0} + 6 c_{1,0} & 6 c_{0,0} + c_{1,0} & c_{0,0} &  &  \\
1001 c_{0,0} + 429 c_{1,0} & 1001 c_{0,0} + 429 c_{1,0} & 704 c_{0,0} + 297 c_{1,0} & 407 c_{0,0} + 165 c_{1,0} & 200 c_{0,0} + 75 c_{1,0} & 83 c_{0,0} + 27 c_{1,0} & 28 c_{0,0} + 7 c_{1,0} & 7 c_{0,0} + c_{1,0} & c_{0,0} &  \\
3432 c_{0,0} + 1430 c_{1,0} & 3432 c_{0,0} + 1430 c_{1,0} & 2431 c_{0,0} + 1001 c_{1,0} & 1430 c_{0,0} + 572 c_{1,0} & 726 c_{0,0} + 275 c_{1,0} & 319 c_{0,0} + 110 c_{1,0} & 119 c_{0,0} + 35 c_{1,0} & 36 c_{0,0} + 8 c_{1,0} & 8 c_{0,0} + c_{1,0} & c_{0,0} \\
\end{matrix}\right]
\label{eq:clean:two:splitted:catalan}
%$}
\end{align}
\begin{equation}
\left[\begin{matrix}
\textcolor{blue}{c_{0,0}} &  &  &  &  &  &  &  &  &  \\
\textcolor{blue}{c_{1,0}} & \textcolor{red}{c_{1,1}} &  &  &  &  &  &  &  &  \\
\textcolor{blue}{c_{2,0}} & \textcolor{red}{c_{2,1}} & \textcolor{red}{c_{2,2}} &  &  &  &  &  &  &  \\
c_{2,0} + c_{2,1} + c_{2,2} & c_{2,0} + c_{2,1} + c_{2,2} & c_{2,1} + c_{2,2} & c_{2,2} &  &  &  &  &  &  \\
2 c_{2,0} + 3 c_{2,1} + 4 c_{2,2} & 2 c_{2,0} + 3 c_{2,1} + 4 c_{2,2} & c_{2,0} + 2 c_{2,1} + 3 c_{2,2} & c_{2,1} + 2 c_{2,2} & c_{2,2} &  &  &  &  &  \\
5 c_{2,0} + 9 c_{2,1} + 14 c_{2,2} & 5 c_{2,0} + 9 c_{2,1} + 14 c_{2,2} & 3 c_{2,0} + 6 c_{2,1} + 10 c_{2,2} & c_{2,0} + 3 c_{2,1} + 6 c_{2,2} & c_{2,1} + 3 c_{2,2} & c_{2,2} &  &  &  &  \\
14 c_{2,0} + 28 c_{2,1} + 48 c_{2,2} & 14 c_{2,0} + 28 c_{2,1} + 48 c_{2,2} & 9 c_{2,0} + 19 c_{2,1} + 34 c_{2,2} & 4 c_{2,0} + 10 c_{2,1} + 20 c_{2,2} & c_{2,0} + 4 c_{2,1} + 10 c_{2,2} & c_{2,1} + 4 c_{2,2} & c_{2,2} &  &  &  \\
42 c_{2,0} + 90 c_{2,1} + 165 c_{2,2} & 42 c_{2,0} + 90 c_{2,1} + 165 c_{2,2} & 28 c_{2,0} + 62 c_{2,1} + 117 c_{2,2} & 14 c_{2,0} + 34 c_{2,1} + 69 c_{2,2} & 5 c_{2,0} + 15 c_{2,1} + 35 c_{2,2} & c_{2,0} + 5 c_{2,1} + 15 c_{2,2} & c_{2,1} + 5 c_{2,2} & c_{2,2} &  &  \\
132 c_{2,0} + 297 c_{2,1} + 572 c_{2,2} & 132 c_{2,0} + 297 c_{2,1} + 572 c_{2,2} & 90 c_{2,0} + 207 c_{2,1} + 407 c_{2,2} & 48 c_{2,0} + 117 c_{2,1} + 242 c_{2,2} & 20 c_{2,0} + 55 c_{2,1} + 125 c_{2,2} & 6 c_{2,0} + 21 c_{2,1} + 56 c_{2,2} & c_{2,0} + 6 c_{2,1} + 21 c_{2,2} & c_{2,1} + 6 c_{2,2} & c_{2,2} &  \\
429 c_{2,0} + 1001 c_{2,1} + 2002 c_{2,2} & 429 c_{2,0} + 1001 c_{2,1} + 2002 c_{2,2} & 297 c_{2,0} + 704 c_{2,1} + 1430 c_{2,2} & 165 c_{2,0} + 407 c_{2,1} + 858 c_{2,2} & 75 c_{2,0} + 200 c_{2,1} + 451 c_{2,2} & 27 c_{2,0} + 83 c_{2,1} + 209 c_{2,2} & 7 c_{2,0} + 28 c_{2,1} + 84 c_{2,2} & c_{2,0} + 7 c_{2,1} + 28 c_{2,2} & c_{2,1} + 7 c_{2,2} & c_{2,2} \\
\end{matrix}\right]
\label{eq:three:splitted:catalan}
\end{equation}
\begin{equation}
\left[\begin{matrix}
\textcolor{blue}{c_{0,0}} &  &  &  &  &  &  &  &  &  \\
\textcolor{blue}{c_{1,0}} & c_{0,0} &  &  &  &  &  &  &  &  \\
\textcolor{blue}{c_{2,0}} & c_{0,0} + c_{1,0} & c_{0,0} &  &  &  &  &  &  &  \\
2 c_{0,0} + c_{1,0} + c_{2,0} & 2 c_{0,0} + c_{1,0} + c_{2,0} & 2 c_{0,0} + c_{1,0} & c_{0,0} &  &  &  &  &  &  \\
7 c_{0,0} + 3 c_{1,0} + 2 c_{2,0} & 7 c_{0,0} + 3 c_{1,0} + 2 c_{2,0} & 5 c_{0,0} + 2 c_{1,0} + c_{2,0} & 3 c_{0,0} + c_{1,0} & c_{0,0} &  &  &  &  &  \\
23 c_{0,0} + 9 c_{1,0} + 5 c_{2,0} & 23 c_{0,0} + 9 c_{1,0} + 5 c_{2,0} & 16 c_{0,0} + 6 c_{1,0} + 3 c_{2,0} & 9 c_{0,0} + 3 c_{1,0} + c_{2,0} & 4 c_{0,0} + c_{1,0} & c_{0,0} &  &  &  &  \\
76 c_{0,0} + 28 c_{1,0} + 14 c_{2,0} & 76 c_{0,0} + 28 c_{1,0} + 14 c_{2,0} & 53 c_{0,0} + 19 c_{1,0} + 9 c_{2,0} & 30 c_{0,0} + 10 c_{1,0} + 4 c_{2,0} & 14 c_{0,0} + 4 c_{1,0} + c_{2,0} & 5 c_{0,0} + c_{1,0} & c_{0,0} &  &  &  \\
255 c_{0,0} + 90 c_{1,0} + 42 c_{2,0} & 255 c_{0,0} + 90 c_{1,0} + 42 c_{2,0} & 179 c_{0,0} + 62 c_{1,0} + 28 c_{2,0} & 103 c_{0,0} + 34 c_{1,0} + 14 c_{2,0} & 50 c_{0,0} + 15 c_{1,0} + 5 c_{2,0} & 20 c_{0,0} + 5 c_{1,0} + c_{2,0} & 6 c_{0,0} + c_{1,0} & c_{0,0} &  &  \\
869 c_{0,0} + 297 c_{1,0} + 132 c_{2,0} & 869 c_{0,0} + 297 c_{1,0} + 132 c_{2,0} & 614 c_{0,0} + 207 c_{1,0} + 90 c_{2,0} & 359 c_{0,0} + 117 c_{1,0} + 48 c_{2,0} & 180 c_{0,0} + 55 c_{1,0} + 20 c_{2,0} & 77 c_{0,0} + 21 c_{1,0} + 6 c_{2,0} & 27 c_{0,0} + 6 c_{1,0} + c_{2,0} & 7 c_{0,0} + c_{1,0} & c_{0,0} &  \\
3003 c_{0,0} + 1001 c_{1,0} + 429 c_{2,0} & 3003 c_{0,0} + 1001 c_{1,0} + 429 c_{2,0} & 2134 c_{0,0} + 704 c_{1,0} + 297 c_{2,0} & 1265 c_{0,0} + 407 c_{1,0} + 165 c_{2,0} & 651 c_{0,0} + 200 c_{1,0} + 75 c_{2,0} & 292 c_{0,0} + 83 c_{1,0} + 27 c_{2,0} & 112 c_{0,0} + 28 c_{1,0} + 7 c_{2,0} & 35 c_{0,0} + 7 c_{1,0} + c_{2,0} & 8 c_{0,0} + c_{1,0} & c_{0,0} \\
\end{matrix}\right]
\label{eq:clean:three:splitted:catalan}
\end{equation}
\end{sidewaystable}
Two-way unfolding suggests that we can characterize coefficients lying on a column $k$ as a
combination of \emph{two} generating functions $a_{k}(t)$ and $b_{k}(t)$, multiplied by $c_{0,0}$
and $c_{1,0}$ respectively. Because coefficients lying on a column $k$ of a Riordan array
can also be characterized by the convolution $d(t)h(t)^{k}$, we can state the following relation:
\begin{equation}
    d(t)h(t)^{k} = c_{0,0}a_{k}(t) + c_{1,0}b_{k}(t)
    \label{eq:two:splitted:catalan:generic:column:relation}
\end{equation}
which can be used toward to assert a relation about two adjacent columns $k$ and $k+1$.
To see this, starts by taking the ratio of those columns:
\begin{displaymath}
    \frac{d(t)h(t)^{k+1}}{d(t)h(t)^{k}} = \frac{c_{0,0}a_{k+1}(t) + c_{1,0}b_{k+1}(t)}
        {c_{0,0}a_{k}(t) + c_{1,0}b_{k}(t)}
\end{displaymath}
which can be simplified to:
\begin{displaymath}
    h(t)\left(c_{0,0}a_{k}(t) + c_{1,0}b_{k}(t)\right) =
        c_{0,0}a_{k+1}(t) + c_{1,0}b_{k+1}(t)
\end{displaymath}
which holds if and only if:
\begin{displaymath}
    h(t) a_{k}(t) = a_{k+1}(t) \quad \wedge \quad h(t) b_{k}(t) = b_{k+1}(t)
\end{displaymath}
unfolding such relations we get, for a generic column $j$ and an integer $s\in\lbrace 0,\ldots,j\rbrace$:
\begin{displaymath}
    a_{j}(t) = h(t)^{j-s} a_{s}(t) \quad \quad
    b_{j}(t) = h(t)^{j-s} b_{s}(t)
\end{displaymath}
fixing $s=0$ we go down to the very first column:
\begin{displaymath}
    a_{j}(t) = h(t)^{j} a_{0}(t) \quad \quad
    b_{j}(t) = h(t)^{j} b_{0}(t)
\end{displaymath}
therefore there exists two Riordan arrays $\mathcal{C}_{a}$ and $\mathcal{C}_{b}$
defined as follows:
\begin{displaymath}
    \mathcal{C}_{a} = \left(a_{0}(t), h(t)\right) \quad \quad
    \mathcal{C}_{b} = \left(b_{0}(t), h(t)\right) \quad \quad
\end{displaymath}
such that:
\begin{equation}
    \mathcal{C} = c_{0,0}\mathcal{C}_{a} + c_{1,0}\mathcal{C}_{b}
    \label{eq:two:splitted:catalan:riordan:expansion}
\end{equation}
From now on, we attach a super-script $^{(i)}$, for $i\in\lbrace 2,3\ldots\rbrace$,
to functions used in a sum expansion in order to be clear that such functions
refers to a matrix unfolding where $i$ variables, namely $c_{0,0},\ldots,c_{i-1,0}$,
are present. In the case above, $a_{0}^{(2)}(t)$ and $b_{0}^{(2)}(t)$
denote the same functions as $a_{0}(t)$ and $b_{0}(t)$.

Studying \autoref{eq:two:splitted:catalan:generic:column:relation}
with $k=0$ we get:
\begin{equation}
    d(t) = c_{0,0}a_{0}^{(2)}(t) + c_{1,0}b_{0}^{(2)}(t)
\end{equation}
a look at \autoref{eq:standard:catalan} allows us to set both $c_{0,0}$ and $c_{1,0}$ to $1$,
and looking at \autoref{eq:clean:two:splitted:catalan} it seems to be permitted to set
function $b_{0}^{(2)}(t)$ to $t\,d(t)$; therefore,
\begin{equation}
    d(t) = a_{0}^{(2)}(t) + t\,d(t) \rightarrow d(t)(1-t) = a_{0}^{(2)}(t)
\end{equation}
holds. So \autoref{eq:two:splitted:catalan:riordan:expansion} can be rewritten as follows:
\begin{displaymath}
    \mathcal{C}(d(t), h(t)) = \mathcal{C}_{a}(d(t)(1-t), h(t)) + \mathcal{C}_{b}(t\,d(t), h(t))
\end{displaymath}
in expanded matrix notation:
\begin{equation}
\hspace{-3cm}
\mathcal{C} =
\left[\begin{matrix}
1 &  &  &  &  &  &  &  &  &  \\
0 & 1 &  &  &  &  &  &  &  &  \\
1 & 1 & 1 &  &  &  &  &  &  &  \\
3 & 3 & 2 & 1 &  &  &  &  &  &  \\
9 & 9 & 6 & 3 & 1 &  &  &  &  &  \\
28 & 28 & 19 & 10 & 4 & 1 &  &  &  &  \\
90 & 90 & 62 & 34 & 15 & 5 & 1 &  &  &  \\
297 & 297 & 207 & 117 & 55 & 21 & 6 & 1 &  &  \\
1001 & 1001 & 704 & 407 & 200 & 83 & 28 & 7 & 1 &  \\
3432 & 3432 & 2431 & 1430 & 726 & 319 & 119 & 36 & 8 & 1 \\
\end{matrix}\right] +
\left[\begin{matrix}
0 &  &  &  &  &  &  &  &  &  \\
1 & 0 &  &  &  &  &  &  &  &  \\
1 & 1 & 0 &  &  &  &  &  &  &  \\
2 & 2 & 1 & 0 &  &  &  &  &  &  \\
5 & 5 & 3 & 1 & 0 &  &  &  &  &  \\
14 & 14 & 9 & 4 & 1 & 0 &  &  &  &  \\
42 & 42 & 28 & 14 & 5 & 1 & 0 &  &  &  \\
132 & 132 & 90 & 48 & 20 & 6 & 1 & 0 &  &  \\
429 & 429 & 297 & 165 & 75 & 27 & 7 & 1 & 0 &  \\
1430 & 1430 & 1001 & 572 & 275 & 110 & 35 & 8 & 1 & 0 \\
\end{matrix}\right]\\
\label{eq:matrix:expansion:two:splitted:catalan}
\end{equation}

\subsection{Three-way splitting}

Now we're going to unfold matrix in \autoref{eq:purely:generic:catalan} starting from row $3$,
aiming to rewrite that matrix such that it depends only on $c_{0,0}, c_{1,0}, c_{2,0}$.
We can characterize the convolution for the generic column $k$ using three
functions $a_{k}^{(3)}(t), b_{k}^{(3)}(t), c_{k}^{(3)}(t)$ as follows:
\begin{equation}
    d(t)h(t)^{k} = c_{0,0}a_{k}^{(3)}(t) + c_{1,0}b_{k}^{(3)}(t) + c_{2,0}c_{k}^{(3)}(t)
    \label{eq:three:splitted:catalan:generic:column:relation}
\end{equation}
In the style of the argument shown in the previous section, we seek for:
\begin{equation}
    \mathcal{C} = c_{0,0}\mathcal{C}_{a} + c_{1,0}\mathcal{C}_{b} + c_{2,0}\mathcal{C}_{c}
    \label{eq:three:splitted:catalan:riordan:expansion}
\end{equation}
where:
\begin{displaymath}
    \mathcal{C}_{a} = \left(a_{0}^{(3)}(t), h(t)\right) \quad \quad
    \mathcal{C}_{b} = \left(b_{0}^{(3)}(t), h(t)\right) \quad \quad
    \mathcal{C}_{c} = \left(c_{0}^{(3)}(t), h(t)\right) \quad \quad
\end{displaymath}
Looking at \autoref{eq:clean:three:splitted:catalan} it seems that the following relations hold:
\begin{displaymath}
    b_{0}^{(3)}(t) = t\,a_{0}^{(2)}(t) = t(1-t)d(t)\quad \quad
    c_{0}^{(3)}(t) = t\,b_{0}^{(2)}(t) = t^{2}d(t)
\end{displaymath}
and \autoref{eq:standard:catalan} allows us to set $c_{2,0}$ to $2$. Sustituting in
\autoref{eq:three:splitted:catalan:generic:column:relation}, with $k=0$, yields
function $a_{0}^{(3)}(t)$:
\begin{displaymath}
    d(t)\left(1 -t(1-t) -2t^{2}\right) = a_{0}^{(3)}(t)
\end{displaymath}
so:
\begin{displaymath}
    d(t)\left(1 -t -t^{2}\right) = a_{0}^{(3)}(t)
\end{displaymath}
therefore $\mathcal{C}$ factorization, using $3$ variables, can be written as:
\begin{displaymath}
    \mathcal{C} =
        \left(\left(1 -t -t^{2}\right)d(t), h(t)\right) +
        \left(\left(t-t^{2}\right)d(t), h(t)\right) +
        2\,\left(t^{2}d(t), h(t)\right)
\end{displaymath}
in matrix notation:
\begin{displaymath}
\hspace{-3cm}
\scriptsize
\mathcal{C} =
\left[\begin{matrix}
1 &  &  &  &  &  &  &  &  &  \\
0 & 1 &  &  &  &  &  &  &  &  \\
0 & 1 & 1 &  &  &  &  &  &  &  \\
2 & 2 & 2 & 1 &  &  &  &  &  &  \\
7 & 7 & 5 & 3 & 1 &  &  &  &  &  \\
23 & 23 & 16 & 9 & 4 & 1 &  &  &  &  \\
76 & 76 & 53 & 30 & 14 & 5 & 1 &  &  &  \\
255 & 255 & 179 & 103 & 50 & 20 & 6 & 1 &  &  \\
869 & 869 & 614 & 359 & 180 & 77 & 27 & 7 & 1 &  \\
3003 & 3003 & 2134 & 1265 & 651 & 292 & 112 & 35 & 8 & 1 \\
\end{matrix}\right] +
\left[\begin{matrix}
0 &  &  &  &  &  &  &  &  &  \\
1 & 0 &  &  &  &  &  &  &  &  \\
0 & 1 & 0 &  &  &  &  &  &  &  \\
1 & 1 & 1 & 0 &  &  &  &  &  &  \\
3 & 3 & 2 & 1 & 0 &  &  &  &  &  \\
9 & 9 & 6 & 3 & 1 & 0 &  &  &  &  \\
28 & 28 & 19 & 10 & 4 & 1 & 0 &  &  &  \\
90 & 90 & 62 & 34 & 15 & 5 & 1 & 0 &  &  \\
297 & 297 & 207 & 117 & 55 & 21 & 6 & 1 & 0 &  \\
1001 & 1001 & 704 & 407 & 200 & 83 & 28 & 7 & 1 & 0 \\
\end{matrix}\right]
\end{displaymath}
\begin{displaymath}
\scriptsize
+ 2\,\left[\begin{matrix}
0 &  &  &  &  &  &  &  &  &  \\
0 & 0 &  &  &  &  &  &  &  &  \\
1 & 0 & 0 &  &  &  &  &  &  &  \\
1 & 1 & 0 & 0 &  &  &  &  &  &  \\
2 & 2 & 1 & 0 & 0 &  &  &  &  &  \\
5 & 5 & 3 & 1 & 0 & 0 &  &  &  &  \\
14 & 14 & 9 & 4 & 1 & 0 & 0 &  &  &  \\
42 & 42 & 28 & 14 & 5 & 1 & 0 & 0 &  &  \\
132 & 132 & 90 & 48 & 20 & 6 & 1 & 0 & 0 &  \\
429 & 429 & 297 & 165 & 75 & 27 & 7 & 1 & 0 & 0 \\
\end{matrix}\right]
\end{displaymath}

\subsection{The Fibonacci triangle, which isn't a renewal array}

In this section we study the Fibonacci triangle, denoted by the following
Riordan array:
\begin{displaymath}
    \mathcal{F}=\left(\frac{1}{1-t-t^{2}}, \frac{1-\sqrt{1-4\,t}}{2}\right)
\end{displaymath}
and with the following matrix expansion:
\begin{equation}
f_{0,0}\left[\begin{array}{cccccccccc}
1 &  &  &  &  &  &  &  &  &  \\
1 & 1 &  &  &  &  &  &  &  &  \\
2 & 2 & 1 &  &  &  &  &  &  &  \\
3 & 5 & 3 & 1 &  &  &  &  &  &  \\
5 & 12 & 9 & 4 & 1 &  &  &  &  &  \\
8 & 31 & 26 & 14 & 5 & 1 &  &  &  &  \\
13 & 85 & 77 & 46 & 20 & 6 & 1 &  &  &  \\
21 & 248 & 235 & 150 & 73 & 27 & 7 & 1 &  &  \\
34 & 762 & 741 & 493 & 258 & 108 & 35 & 8 & 1 &  \\
55 & 2440 & 2406 & 1644 & 903 & 410 & 152 & 44 & 9 & 1 \\
\end{array}\right]
\end{equation}
produced by a purely symbolic matrix as the one in \autoref{eq:purely:generic:catalan},
changing the generic symbol from $c$ to $f$, according $A$-sequence:
\begin{displaymath}
    A(t)=\frac{1}{1-t}
\end{displaymath}
and $Z$-sequence:
\begin{displaymath}
    Z(t)=1+t-t^{2}
\end{displaymath}

In \autoref{eq:fibonacci:four:splitted} we report a splitting using $4$ free variables
and in \autoref{eq:fibonacci:four:splitted:matrix:expansion} the corresponding matrix expansion.
\begin{sidewaystable}
\scriptsize
\begin{equation}
\left[\begin{array}{cccccccccc}
\textcolor{blue}{f_{0,0}} &  &  &  &  &  &  &  &  &  \\
\textcolor{blue}{f_{1,0}} & f_{0,0} &  &  &  &  &  &  &  &  \\
\textcolor{blue}{f_{2,0}} & f_{0,0} + f_{1,0} & f_{0,0} &  &  &  &  &  &  &  \\
\textcolor{blue}{f_{3,0}} & 2 f_{0,0} + f_{1,0} + f_{2,0} & 2 f_{0,0} + f_{1,0} & f_{0,0} &  &  &  &  &  &  \\
f_{2,0} + f_{3,0} & 5 f_{0,0} + 2 f_{1,0} + f_{2,0} + f_{3,0} & 5 f_{0,0} + 2 f_{1,0} + f_{2,0} & 3 f_{0,0} + f_{1,0} & f_{0,0} &  &  &  &  &  \\
f_{2,0} + 2 f_{3,0} & 14 f_{0,0} + 5 f_{1,0} + 3 f_{2,0} + 2 f_{3,0} & 14 f_{0,0} + 5 f_{1,0} + 2 f_{2,0} + f_{3,0} & 9 f_{0,0} + 3 f_{1,0} + f_{2,0} & 4 f_{0,0} + f_{1,0} & f_{0,0} &  &  &  &  \\
2 f_{2,0} + 3 f_{3,0} & 42 f_{0,0} + 14 f_{1,0} + 7 f_{2,0} + 5 f_{3,0} & 42 f_{0,0} + 14 f_{1,0} + 6 f_{2,0} + 3 f_{3,0} & 28 f_{0,0} + 9 f_{1,0} + 3 f_{2,0} + f_{3,0} & 14 f_{0,0} + 4 f_{1,0} + f_{2,0} & 5 f_{0,0} + f_{1,0} & f_{0,0} &  &  &  \\
3 f_{2,0} + 5 f_{3,0} & 132 f_{0,0} + 42 f_{1,0} + 19 f_{2,0} + 12 f_{3,0} & 132 f_{0,0} + 42 f_{1,0} + 17 f_{2,0} + 9 f_{3,0} & 90 f_{0,0} + 28 f_{1,0} + 10 f_{2,0} + 4 f_{3,0} & 48 f_{0,0} + 14 f_{1,0} + 4 f_{2,0} + f_{3,0} & 20 f_{0,0} + 5 f_{1,0} + f_{2,0} & 6 f_{0,0} + f_{1,0} & f_{0,0} &  &  \\
5 f_{2,0} + 8 f_{3,0} & 429 f_{0,0} + 132 f_{1,0} + 54 f_{2,0} + 31 f_{3,0} & 429 f_{0,0} + 132 f_{1,0} + 51 f_{2,0} + 26 f_{3,0} & 297 f_{0,0} + 90 f_{1,0} + 32 f_{2,0} + 14 f_{3,0} & 165 f_{0,0} + 48 f_{1,0} + 15 f_{2,0} + 5 f_{3,0} & 75 f_{0,0} + 20 f_{1,0} + 5 f_{2,0} + f_{3,0} & 27 f_{0,0} + 6 f_{1,0} + f_{2,0} & 7 f_{0,0} + f_{1,0} & f_{0,0} &  \\
8 f_{2,0} + 13 f_{3,0} & 1430 f_{0,0} + 429 f_{1,0} + 163 f_{2,0} + 85 f_{3,0} & 1430 f_{0,0} + 429 f_{1,0} + 158 f_{2,0} + 77 f_{3,0} & 1001 f_{0,0} + 297 f_{1,0} + 104 f_{2,0} + 46 f_{3,0} & 572 f_{0,0} + 165 f_{1,0} + 53 f_{2,0} + 20 f_{3,0} & 275 f_{0,0} + 75 f_{1,0} + 21 f_{2,0} + 6 f_{3,0} & 110 f_{0,0} + 27 f_{1,0} + 6 f_{2,0} + f_{3,0} & 35 f_{0,0} + 7 f_{1,0} + f_{2,0} & 8 f_{0,0} + f_{1,0} & f_{0,0} \\
\end{array}\right]
\label{eq:fibonacci:four:splitted}
\end{equation}
\begin{equation}
\begin{split}
& f_{0,0}\left[\begin{array}{cccccccccc}
1 &  &  &  &  &  &  &  &  &  \\
0 & 1 &  &  &  &  &  &  &  &  \\
0 & 1 & 1 &  &  &  &  &  &  &  \\
0 & 2 & 2 & 1 &  &  &  &  &  &  \\
0 & 5 & 5 & 3 & 1 &  &  &  &  &  \\
0 & 14 & 14 & 9 & 4 & 1 &  &  &  &  \\
0 & 42 & 42 & 28 & 14 & 5 & 1 &  &  &  \\
0 & 132 & 132 & 90 & 48 & 20 & 6 & 1 &  &  \\
0 & 429 & 429 & 297 & 165 & 75 & 27 & 7 & 1 &  \\
0 & 1430 & 1430 & 1001 & 572 & 275 & 110 & 35 & 8 & 1 \\
\end{array}\right] + f_{1,0}\left[\begin{array}{cccccccccc}
0 &  &  &  &  &  &  &  &  &  \\
1 & 0 &  &  &  &  &  &  &  &  \\
0 & 1 & 0 &  &  &  &  &  &  &  \\
0 & 1 & 1 & 0 &  &  &  &  &  &  \\
0 & 2 & 2 & 1 & 0 &  &  &  &  &  \\
0 & 5 & 5 & 3 & 1 & 0 &  &  &  &  \\
0 & 14 & 14 & 9 & 4 & 1 & 0 &  &  &  \\
0 & 42 & 42 & 28 & 14 & 5 & 1 & 0 &  &  \\
0 & 132 & 132 & 90 & 48 & 20 & 6 & 1 & 0 &  \\
0 & 429 & 429 & 297 & 165 & 75 & 27 & 7 & 1 & 0 \\
\end{array}\right] \\
& + f_{2,0}\left[\begin{array}{cccccccccc}
0 &  &  &  &  &  &  &  &  &  \\
0 & 0 &  &  &  &  &  &  &  &  \\
1 & 0 & 0 &  &  &  &  &  &  &  \\
0 & 1 & 0 & 0 &  &  &  &  &  &  \\
1 & 1 & 1 & 0 & 0 &  &  &  &  &  \\
1 & 3 & 2 & 1 & 0 & 0 &  &  &  &  \\
2 & 7 & 6 & 3 & 1 & 0 & 0 &  &  &  \\
3 & 19 & 17 & 10 & 4 & 1 & 0 & 0 &  &  \\
5 & 54 & 51 & 32 & 15 & 5 & 1 & 0 & 0 &  \\
8 & 163 & 158 & 104 & 53 & 21 & 6 & 1 & 0 & 0 \\
\end{array}\right] + f_{3,0}\left[\begin{array}{cccccccccc}
0 &  &  &  &  &  &  &  &  &  \\
0 & 0 &  &  &  &  &  &  &  &  \\
0 & 0 & 0 &  &  &  &  &  &  &  \\
1 & 0 & 0 & 0 &  &  &  &  &  &  \\
1 & 1 & 0 & 0 & 0 &  &  &  &  &  \\
2 & 2 & 1 & 0 & 0 & 0 &  &  &  &  \\
3 & 5 & 3 & 1 & 0 & 0 & 0 &  &  &  \\
5 & 12 & 9 & 4 & 1 & 0 & 0 & 0 &  &  \\
8 & 31 & 26 & 14 & 5 & 1 & 0 & 0 & 0 &  \\
13 & 85 & 77 & 46 & 20 & 6 & 1 & 0 & 0 & 0 \\
\end{array}\right]
\end{split}
\label{eq:fibonacci:four:splitted:matrix:expansion}
\end{equation}
\end{sidewaystable}

\subsection{A possible generalization}

Arguments in previous sections, were unfolding leaving $2$ and $3$ variables are
described, allow us to conjecture a general factorization for a Riordan array $\mathcal{C}$.
Let $\left\lbrace \alpha_{k,i}^{(s)}(t)\right\rbrace_{k,i\in\mathbb{N}}$ be a family of generating functions
in the indeterminate $t$, related to matrix unfolding depending on $s$ free variables.
Then we can state the following relation about the convolution characterizing a
generic column $k$ of $\mathcal{C}$ as follows:
\begin{equation}
    d(t)h(t)^{k} = c_{0,0}\alpha_{k,0}^{(s)}(t) + c_{1,0}\alpha_{k,1}^{(s)}(t) + \ldots
        c_{s-1,0}\alpha_{k,s-1}^{(s)}(t)
\end{equation}
setting $k=0$ we can get rid of function $h$, looking for a factorization of function $d$:
\begin{equation}
    d(t) = c_{0,0}\alpha_{0,0}^{(s)}(t) + c_{1,0}\alpha_{0,1}^{(s)}(t) + \ldots +
        c_{s-1,0}\alpha_{0,s-1}^{(s)}(t)
\end{equation}
where $\alpha_{0, j}^{(s)}(t) = t\,\alpha_{0, j-1}^{(s-1)}(t)$, for $j\in\lbrace 1,\ldots,s-1\rbrace$;
the base case for the recursion is $\alpha_{0, 1}^{(2)}(t) = t\,d(t)$.

In \autoref{eq:table:a:zero:zero:pascal}, \autoref{eq:table:a:zero:zero:shapiro},
\autoref{eq:table:a:zero:zero:catalan} and \autoref{eq:table:a:zero:zero:fibonacci},
we report three tables with schemata to compute
$\alpha_{0,0}^{(s)}(t)$, for $s\in\lbrace 2, 3, 4, 5\rbrace$, relative to
Pascal triangle $\mathcal{P}$, Shapiro triangle $\mathcal{S}$,
Catalan triangle $\mathcal{C}$ and, finally, Fibonacci triangle $\mathcal{F}$, respectively.
Recall the \emph{fundamental}
base case for the recursion $\alpha_{0,1}^{(2)}(t) = t\,d(t)$, then the following
definitions for $d(t)$ functions and initial coefficients are used for each array:
\begin{itemize}
\item $\mathcal{P}$:
\begin{displaymath}
     d(t) = \frac{1}{1-t} \quad \quad p_{i,0}=1, i\in\lbrace 0,\ldots,4\rbrace
\end{displaymath}

\item $\mathcal{S}$:
\begin{displaymath}
        d(t) = \frac{1-2t-\sqrt{1-4t}}{2t^{2}} \quad \quad
            s_{0,0}=1\quad s_{1,0}=2\quad s_{2,0}=5\quad s_{3,0}=14\quad s_{4,0}=42
\end{displaymath}

\item $\mathcal{C}$:
\begin{displaymath}
        d(t) = \frac{1-\sqrt{1-4t}}{2t} \quad \quad
            c_{0,0}=1\quad c_{1,0}=1\quad c_{2,0}=2\quad c_{3,0}=5\quad c_{4,0}=14
\end{displaymath}

\item $\mathcal{F}$:
\begin{displaymath}
        d(t) = \frac{1}{1-t-t^{2}} \quad \quad
            f_{0,0}=1\quad f_{1,0}=1\quad f_{2,0}=2\quad f_{3,0}=3\quad f_{4,0}=5
\end{displaymath}

\end{itemize}

\begin{sidewaystable}
\scriptsize

% Pascal triangle
\begin{equation}
    %\hspace{-3cm}
    \begin{array}{ccc}
        s & d(t) & \alpha_{0,0}^{(s)}(t) \\
        \hline
        2 & p_{0,0}\alpha_{0,0}^{(2)}(t) + p_{1,0}\alpha_{0,1}^{(2)}(t) = \alpha_{0,0}^{(2)}(t) + t\,d(t) & 1 \\
        3 & p_{0,0}\alpha_{0,0}^{(3)}(t) + t\left(p_{1,0}\alpha_{0,0}^{(2)}(t) + p_{2,0}\alpha_{0,1}^{(2)}(t)\right) =
            \alpha_{0,0}^{(3)}(t) + t + t^{2}\,d(t) & 1 \\
        4 & p_{0,0}\alpha_{0,0}^{(4)}(t) + t\left(p_{1,0}\alpha_{0,0}^{(3)}(t) + p_{2,0}\alpha_{0,1}^{(3)}(t) + p_{3,0}\alpha_{0,2}^{(3)}(t)\right)  =
            \alpha_{0,0}^{(4)}(t) + t + t^{2} + t^{3}\,d(t) & 1 \\
        5 & p_{0,0}\alpha_{0,0}^{(5)}(t) + t\left(p_{1,0}\alpha_{0,0}^{(4)}(t) + p_{2,0}\alpha_{0,1}^{(4)}(t) + p_{3,0}\alpha_{0,2}^{(4)}(t) + p_{4,0}\alpha_{0,3}^{(4)}(t)\right)  =
            \alpha_{0,0}^{(5)}(t) + t + t^{2} + t^{3} + t^{4}\,d(t) & 1 \\
    \end{array}
    \label{eq:table:a:zero:zero:pascal}
\end{equation}

% Shapiro triangle
\begin{equation}
    %\hspace{-5cm}
    \begin{array}{ccc}
        s & d(t) & \alpha_{0,0}^{(s)}(t) \\
        \hline
        2 & s_{0,0}\alpha_{0,0}^{(2)}(t) + s_{1,0}\alpha_{0,1}^{(2)}(t) = \alpha_{0,0}^{(2)}(t) + 2t\,d(t) & d(t)(1-2t) \\
        3 & s_{0,0}\alpha_{0,0}^{(3)}(t) + t\left(s_{1,0}\alpha_{0,0}^{(2)}(t) + s_{2,0}\alpha_{0,1}^{(2)}(t)\right) =
            \alpha_{0,0}^{(3)}(t) + 2t(1-2t)d(t) + 5t^{2}\,d(t) & d(t)\left(1-2t-t^{2}\right) \\
        4 & s_{0,0}\alpha_{0,0}^{(4)}(t) + t\left(s_{1,0}\alpha_{0,0}^{(3)}(t) + s_{2,0}\alpha_{0,1}^{(3)}(t) + s_{3,0}\alpha_{0,2}^{(3)}(t)\right)  =
            \alpha_{0,0}^{(4)}(t) + 2t\,d(t)\left(1-2t-t^{2}\right) + 5t^{2}\,d(t)(1-2t) + 14t^{3}\,d(t) & d(t)\left(1-2t-t^{2}-2t^{3}\right) \\
        5 & s_{0,0}\alpha_{0,0}^{(5)}(t) + t\left(s_{1,0}\alpha_{0,0}^{(4)}(t) + s_{2,0}\alpha_{0,1}^{(4)}(t) + s_{3,0}\alpha_{0,2}^{(4)}(t) + s_{4,0}\alpha_{0,3}^{(4)}(t)\right)  =
            \alpha_{0,0}^{(5)}(t) + 2t\,d(t)\left(1-2t-t^{2}-2t^{3}\right) + 5t^{2}\,d(t)\left(1-2t-t^{2}\right) + 14t^{3}\,d(t)(1-2t) + 42t^{4}\,d(t) & d(t)\left(1-2t-t^{2}-2t^{3}-5t^{4}\right) \\
    \end{array}
    \label{eq:table:a:zero:zero:shapiro}
\end{equation}

% Catalan triangle
\begin{equation}
    %\hspace{-5cm}
    \begin{array}{ccc}
        s & d(t) & \alpha_{0,0}^{(s)}(t) \\
        \hline
        2 & c_{0,0}\alpha_{0,0}^{(2)}(t) + c_{1,0}\alpha_{0,1}^{(2)}(t) = \alpha_{0,0}^{(2)}(t) + t\,d(t) & d(t)(1-t) \\
        3 & c_{0,0}\alpha_{0,0}^{(3)}(t) + t\left(c_{1,0}\alpha_{0,0}^{(2)}(t) + c_{2,0}\alpha_{0,1}^{(2)}(t)\right) =
            \alpha_{0,0}^{(3)}(t) + t(1-t)d(t) + 2t^{2}\,d(t) & d(t)\left(1-t-t^{2}\right) \\
        4 & c_{0,0}\alpha_{0,0}^{(4)}(t) + t\left(c_{1,0}\alpha_{0,0}^{(3)}(t) + c_{2,0}\alpha_{0,1}^{(3)}(t) + c_{3,0}\alpha_{0,2}^{(3)}(t)\right)  =
            \alpha_{0,0}^{(4)}(t) + t\,d(t)\left(1-t-t^{2}\right) + 2t^{2}\,d(t)(1-t) + 5t^{3}\,d(t) & d(t)\left(1-t-t^{2}-2t^{3}\right) \\
        5 & c_{0,0}\alpha_{0,0}^{(5)}(t) + t\left(c_{1,0}\alpha_{0,0}^{(4)}(t) + c_{2,0}\alpha_{0,1}^{(4)}(t) + c_{3,0}\alpha_{0,2}^{(4)}(t) + c_{4,0}\alpha_{0,3}^{(4)}(t)\right)  =
            \alpha_{0,0}^{(5)}(t) + t\,d(t)\left(1-t-t^{2}-2t^{3}\right) + 2t^{2}\,d(t)\left(1-t-t^{2}\right) + 5t^{3}\,d(t)(1-t) + 14t^{4}\,d(t) & d(t)\left(1-t-t^{2}-2t^{3}-5t^{4}\right) \\
    \end{array}
    \label{eq:table:a:zero:zero:catalan}
\end{equation}

% Fibonacci triangle
\begin{equation}
    %\hspace{-5cm}
    \begin{array}{ccc}
        s & d(t) & \alpha_{0,0}^{(s)}(t) \\
        \hline
        2 & f_{0,0}\alpha_{0,0}^{(2)}(t) + f_{1,0}\alpha_{0,1}^{(2)}(t) = \alpha_{0,0}^{(2)}(t) + t\,d(t) & d(t)(1-t) \\
        3 & f_{0,0}\alpha_{0,0}^{(3)}(t) + t\left(f_{1,0}\alpha_{0,0}^{(2)}(t) + f_{2,0}\alpha_{0,1}^{(2)}(t)\right) =
            \alpha_{0,0}^{(3)}(t) + t(1-t)d(t) + 2t^{2}\,d(t) & 1 \\
        4 & f_{0,0}\alpha_{0,0}^{(4)}(t) + t\left(f_{1,0}\alpha_{0,0}^{(3)}(t) + f_{2,0}\alpha_{0,1}^{(3)}(t) + f_{3,0}\alpha_{0,2}^{(3)}(t)\right)  =
            \alpha_{0,0}^{(4)}(t) + t + 2t^{2}\,d(t)(1-t) + 3t^{3}\,d(t) & 1 \\
        5 & f_{0,0}\alpha_{0,0}^{(5)}(t) + t\left(f_{1,0}\alpha_{0,0}^{(4)}(t) + f_{2,0}\alpha_{0,1}^{(4)}(t) + f_{3,0}\alpha_{0,2}^{(4)}(t) + f_{4,0}\alpha_{0,3}^{(4)}(t)\right)  =
            \alpha_{0,0}^{(5)}(t) + t + 2t^{2} + 3t^{3}\,d(t)(1-t) + 5t^{4}\,d(t) & 1 \\
    \end{array}
    \label{eq:table:a:zero:zero:fibonacci}
\end{equation}

\end{sidewaystable}

\section{An horizontally stretched Pascal-like array}

In this section we study a stretched version of Pascal-like arrays,
splitting a matrix in $2$ and $3$ variables respectively. The
following is the used recurrence to do unfoldings:
\begin{displaymath}
    m_{n + 1,k + 1} =  m_{n,k - 1} + m_{n,k}
\end{displaymath}
starting with the following purely symbolic matrix:
\begin{equation}
\hspace{-4cm}
\scriptsize
\left[\begin{array}{ccccccccccccccccccc}
m_{0,0} &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
m_{1,0} & m_{1,1} & m_{1,2} &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
m_{2,0} & m_{2,1} & m_{2,2} & m_{2,3} & m_{2,4} &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
m_{3,0} & m_{3,1} & m_{3,2} & m_{3,3} & m_{3,4} & m_{3,5} & m_{3,6} &  &  &  &  &  &  &  &  &  &  &  &  \\
m_{4,0} & m_{4,1} & m_{4,2} & m_{4,3} & m_{4,4} & m_{4,5} & m_{4,6} & m_{4,7} & m_{4,8} &  &  &  &  &  &  &  &  &  &  \\
m_{5,0} & m_{5,1} & m_{5,2} & m_{5,3} & m_{5,4} & m_{5,5} & m_{5,6} & m_{5,7} & m_{5,8} & m_{5,9} & m_{5,10} &  &  &  &  &  &  &  &  \\
m_{6,0} & m_{6,1} & m_{6,2} & m_{6,3} & m_{6,4} & m_{6,5} & m_{6,6} & m_{6,7} & m_{6,8} & m_{6,9} & m_{6,10} & m_{6,11} & m_{6,12} &  &  &  &  &  &  \\
m_{7,0} & m_{7,1} & m_{7,2} & m_{7,3} & m_{7,4} & m_{7,5} & m_{7,6} & m_{7,7} & m_{7,8} & m_{7,9} & m_{7,10} & m_{7,11} & m_{7,12} & m_{7,13} & m_{7,14} &  &  &  &  \\
m_{8,0} & m_{8,1} & m_{8,2} & m_{8,3} & m_{8,4} & m_{8,5} & m_{8,6} & m_{8,7} & m_{8,8} & m_{8,9} & m_{8,10} & m_{8,11} & m_{8,12} & m_{8,13} & m_{8,14} & m_{8,15} & m_{8,16} &  &  \\
m_{9,0} & m_{9,1} & m_{9,2} & m_{9,3} & m_{9,4} & m_{9,5} & m_{9,6} & m_{9,7} & m_{9,8} & m_{9,9} & m_{9,10} & m_{9,11} & m_{9,12} & m_{9,13} & m_{9,14} & m_{9,15} & m_{9,16} & m_{9,17} & m_{9,18} \\
\end{array}\right]
\label{eq:purely:symbolic:stretched:pascal:like}
\end{equation}
It should be pointed out that the given recurrence alone cannot characterize the entire
triangle, the \emph{very first column} in particular. Therefore, we need to use another
recurrence for such column, what in the jargon of Riordan arrays is called $Z$-sequence:
\begin{displaymath}
    m_{n + 1,0} =  m_{n,0}
\end{displaymath}

In \autoref{eq:stretched:pascal:like} we report the unfolding starting at row $1$;
moreover, in \autoref{eq:stretched:pascal:like:two:splitted} and
\autoref{eq:stretched:pascal:like:two:splitted:matrix:expansion} we report the same array splitted
using two variables and the corresponding matrix expansion as summation, respectively.
In parallel, \autoref{eq:stretched:pascal:like:three:splitted} and
\autoref{eq:stretched:pascal:like:three:splitted:matrix:expansion} show the
same triangle splitted respect to $3$ variables.
\begin{equation}
m_{0,0}\left[\begin{array}{ccccccccccccccccccc}
1 &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
1 & 1 & 1 &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
1 & 1 & 2 & 2 & 1 &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
1 & 1 & 2 & 3 & 4 & 3 & 1 &  &  &  &  &  &  &  &  &  &  &  &  \\
1 & 1 & 2 & 3 & 5 & 7 & 7 & 4 & 1 &  &  &  &  &  &  &  &  &  &  \\
1 & 1 & 2 & 3 & 5 & 8 & 12 & 14 & 11 & 5 & 1 &  &  &  &  &  &  &  &  \\
1 & 1 & 2 & 3 & 5 & 8 & 13 & 20 & 26 & 25 & 16 & 6 & 1 &  &  &  &  &  &  \\
1 & 1 & 2 & 3 & 5 & 8 & 13 & 21 & 33 & 46 & 51 & 41 & 22 & 7 & 1 &  &  &  &  \\
1 & 1 & 2 & 3 & 5 & 8 & 13 & 21 & 34 & 54 & 79 & 97 & 92 & 63 & 29 & 8 & 1 &  &  \\
1 & 1 & 2 & 3 & 5 & 8 & 13 & 21 & 34 & 55 & 88 & 133 & 176 & 189 & 155 & 92 & 37 & 9 & 1 \\
\end{array}\right]
\label{eq:stretched:pascal:like}
\end{equation}

\begin{sidewaystable}
\scriptsize
\begin{equation}
\left[\begin{array}{ccccccccccccccccccc}
\textcolor{blue}{m_{0,0}} &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
\textcolor{blue}{m_{1,0}} & m_{0,0} & m_{0,0} &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
m_{1,0} & m_{1,0} & m_{0,0} + m_{1,0} & 2 m_{0,0} & m_{0,0} &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
m_{1,0} & m_{1,0} & 2 m_{1,0} & m_{0,0} + 2 m_{1,0} & 3 m_{0,0} + m_{1,0} & 3 m_{0,0} & m_{0,0} &  &  &  &  &  &  &  &  &  &  &  &  \\
m_{1,0} & m_{1,0} & 2 m_{1,0} & 3 m_{1,0} & m_{0,0} + 4 m_{1,0} & 4 m_{0,0} + 3 m_{1,0} & 6 m_{0,0} + m_{1,0} & 4 m_{0,0} & m_{0,0} &  &  &  &  &  &  &  &  &  &  \\
m_{1,0} & m_{1,0} & 2 m_{1,0} & 3 m_{1,0} & 5 m_{1,0} & m_{0,0} + 7 m_{1,0} & 5 m_{0,0} + 7 m_{1,0} & 10 m_{0,0} + 4 m_{1,0} & 10 m_{0,0} + m_{1,0} & 5 m_{0,0} & m_{0,0} &  &  &  &  &  &  &  &  \\
m_{1,0} & m_{1,0} & 2 m_{1,0} & 3 m_{1,0} & 5 m_{1,0} & 8 m_{1,0} & m_{0,0} + 12 m_{1,0} & 6 m_{0,0} + 14 m_{1,0} & 15 m_{0,0} + 11 m_{1,0} & 20 m_{0,0} + 5 m_{1,0} & 15 m_{0,0} + m_{1,0} & 6 m_{0,0} & m_{0,0} &  &  &  &  &  &  \\
m_{1,0} & m_{1,0} & 2 m_{1,0} & 3 m_{1,0} & 5 m_{1,0} & 8 m_{1,0} & 13 m_{1,0} & m_{0,0} + 20 m_{1,0} & 7 m_{0,0} + 26 m_{1,0} & 21 m_{0,0} + 25 m_{1,0} & 35 m_{0,0} + 16 m_{1,0} & 35 m_{0,0} + 6 m_{1,0} & 21 m_{0,0} + m_{1,0} & 7 m_{0,0} & m_{0,0} &  &  &  &  \\
m_{1,0} & m_{1,0} & 2 m_{1,0} & 3 m_{1,0} & 5 m_{1,0} & 8 m_{1,0} & 13 m_{1,0} & 21 m_{1,0} & m_{0,0} + 33 m_{1,0} & 8 m_{0,0} + 46 m_{1,0} & 28 m_{0,0} + 51 m_{1,0} & 56 m_{0,0} + 41 m_{1,0} & 70 m_{0,0} + 22 m_{1,0} & 56 m_{0,0} + 7 m_{1,0} & 28 m_{0,0} + m_{1,0} & 8 m_{0,0} & m_{0,0} &  &  \\
m_{1,0} & m_{1,0} & 2 m_{1,0} & 3 m_{1,0} & 5 m_{1,0} & 8 m_{1,0} & 13 m_{1,0} & 21 m_{1,0} & 34 m_{1,0} & m_{0,0} + 54 m_{1,0} & 9 m_{0,0} + 79 m_{1,0} & 36 m_{0,0} + 97 m_{1,0} & 84 m_{0,0} + 92 m_{1,0} & 126 m_{0,0} + 63 m_{1,0} & 126 m_{0,0} + 29 m_{1,0} & 84 m_{0,0} + 8 m_{1,0} & 36 m_{0,0} + m_{1,0} & 9 m_{0,0} & m_{0,0} \\
\end{array}\right]
\label{eq:stretched:pascal:like:two:splitted}
\end{equation}
\begin{equation}
m_{0,0}\left[\begin{array}{ccccccccccccccccccc}
1 &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
0 & 1 & 1 &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
0 & 0 & 1 & 2 & 1 &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
0 & 0 & 0 & 1 & 3 & 3 & 1 &  &  &  &  &  &  &  &  &  &  &  &  \\
0 & 0 & 0 & 0 & 1 & 4 & 6 & 4 & 1 &  &  &  &  &  &  &  &  &  &  \\
0 & 0 & 0 & 0 & 0 & 1 & 5 & 10 & 10 & 5 & 1 &  &  &  &  &  &  &  &  \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 6 & 15 & 20 & 15 & 6 & 1 &  &  &  &  &  &  \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 7 & 21 & 35 & 35 & 21 & 7 & 1 &  &  &  &  \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 8 & 28 & 56 & 70 & 56 & 28 & 8 & 1 &  &  \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 9 & 36 & 84 & 126 & 126 & 84 & 36 & 9 & 1 \\
\end{array}\right] + m_{1,0}\left[\begin{array}{ccccccccccccccccccc}
0 &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
1 & 0 & 0 &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
1 & 1 & 1 & 0 & 0 &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
1 & 1 & 2 & 2 & 1 & 0 & 0 &  &  &  &  &  &  &  &  &  &  &  &  \\
1 & 1 & 2 & 3 & 4 & 3 & 1 & 0 & 0 &  &  &  &  &  &  &  &  &  &  \\
1 & 1 & 2 & 3 & 5 & 7 & 7 & 4 & 1 & 0 & 0 &  &  &  &  &  &  &  &  \\
1 & 1 & 2 & 3 & 5 & 8 & 12 & 14 & 11 & 5 & 1 & 0 & 0 &  &  &  &  &  &  \\
1 & 1 & 2 & 3 & 5 & 8 & 13 & 20 & 26 & 25 & 16 & 6 & 1 & 0 & 0 &  &  &  &  \\
1 & 1 & 2 & 3 & 5 & 8 & 13 & 21 & 33 & 46 & 51 & 41 & 22 & 7 & 1 & 0 & 0 &  &  \\
1 & 1 & 2 & 3 & 5 & 8 & 13 & 21 & 34 & 54 & 79 & 97 & 92 & 63 & 29 & 8 & 1 & 0 & 0 \\
\end{array}\right]
\label{eq:stretched:pascal:like:two:splitted:matrix:expansion}
\end{equation}
%\end{sidewaystable}
%\begin{sidewaystable}
%\scriptsize
\begin{equation}
\left[\begin{array}{ccccccccccccccccccc}
\textcolor{blue}{m_{0,0}} &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
\textcolor{blue}{m_{1,0}} & m_{0,0} & m_{0,0} &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
\textcolor{blue}{m_{2,0}} & m_{1,0} & m_{0,0} + m_{1,0} & 2 m_{0,0} & m_{0,0} &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
m_{2,0} & m_{2,0} & m_{1,0} + m_{2,0} & m_{0,0} + 2 m_{1,0} & 3 m_{0,0} + m_{1,0} & 3 m_{0,0} & m_{0,0} &  &  &  &  &  &  &  &  &  &  &  &  \\
m_{2,0} & m_{2,0} & 2 m_{2,0} & m_{1,0} + 2 m_{2,0} & m_{0,0} + 3 m_{1,0} + m_{2,0} & 4 m_{0,0} + 3 m_{1,0} & 6 m_{0,0} + m_{1,0} & 4 m_{0,0} & m_{0,0} &  &  &  &  &  &  &  &  &  &  \\
m_{2,0} & m_{2,0} & 2 m_{2,0} & 3 m_{2,0} & m_{1,0} + 4 m_{2,0} & m_{0,0} + 4 m_{1,0} + 3 m_{2,0} & 5 m_{0,0} + 6 m_{1,0} + m_{2,0} & 10 m_{0,0} + 4 m_{1,0} & 10 m_{0,0} + m_{1,0} & 5 m_{0,0} & m_{0,0} &  &  &  &  &  &  &  &  \\
m_{2,0} & m_{2,0} & 2 m_{2,0} & 3 m_{2,0} & 5 m_{2,0} & m_{1,0} + 7 m_{2,0} & m_{0,0} + 5 m_{1,0} + 7 m_{2,0} & 6 m_{0,0} + 10 m_{1,0} + 4 m_{2,0} & 15 m_{0,0} + 10 m_{1,0} + m_{2,0} & 20 m_{0,0} + 5 m_{1,0} & 15 m_{0,0} + m_{1,0} & 6 m_{0,0} & m_{0,0} &  &  &  &  &  &  \\
m_{2,0} & m_{2,0} & 2 m_{2,0} & 3 m_{2,0} & 5 m_{2,0} & 8 m_{2,0} & m_{1,0} + 12 m_{2,0} & m_{0,0} + 6 m_{1,0} + 14 m_{2,0} & 7 m_{0,0} + 15 m_{1,0} + 11 m_{2,0} & 21 m_{0,0} + 20 m_{1,0} + 5 m_{2,0} & 35 m_{0,0} + 15 m_{1,0} + m_{2,0} & 35 m_{0,0} + 6 m_{1,0} & 21 m_{0,0} + m_{1,0} & 7 m_{0,0} & m_{0,0} &  &  &  &  \\
m_{2,0} & m_{2,0} & 2 m_{2,0} & 3 m_{2,0} & 5 m_{2,0} & 8 m_{2,0} & 13 m_{2,0} & m_{1,0} + 20 m_{2,0} & m_{0,0} + 7 m_{1,0} + 26 m_{2,0} & 8 m_{0,0} + 21 m_{1,0} + 25 m_{2,0} & 28 m_{0,0} + 35 m_{1,0} + 16 m_{2,0} & 56 m_{0,0} + 35 m_{1,0} + 6 m_{2,0} & 70 m_{0,0} + 21 m_{1,0} + m_{2,0} & 56 m_{0,0} + 7 m_{1,0} & 28 m_{0,0} + m_{1,0} & 8 m_{0,0} & m_{0,0} &  &  \\
m_{2,0} & m_{2,0} & 2 m_{2,0} & 3 m_{2,0} & 5 m_{2,0} & 8 m_{2,0} & 13 m_{2,0} & 21 m_{2,0} & m_{1,0} + 33 m_{2,0} & m_{0,0} + 8 m_{1,0} + 46 m_{2,0} & 9 m_{0,0} + 28 m_{1,0} + 51 m_{2,0} & 36 m_{0,0} + 56 m_{1,0} + 41 m_{2,0} & 84 m_{0,0} + 70 m_{1,0} + 22 m_{2,0} & 126 m_{0,0} + 56 m_{1,0} + 7 m_{2,0} & 126 m_{0,0} + 28 m_{1,0} + m_{2,0} & 84 m_{0,0} + 8 m_{1,0} & 36 m_{0,0} + m_{1,0} & 9 m_{0,0} & m_{0,0} \\
\end{array}\right]
\label{eq:stretched:pascal:like:three:splitted}
\end{equation}
\begin{equation}
\begin{split}
& m_{0,0}\left[\begin{array}{ccccccccccccccccccc}
1 &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
0 & 1 & 1 &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
0 & 0 & 1 & 2 & 1 &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
0 & 0 & 0 & 1 & 3 & 3 & 1 &  &  &  &  &  &  &  &  &  &  &  &  \\
0 & 0 & 0 & 0 & 1 & 4 & 6 & 4 & 1 &  &  &  &  &  &  &  &  &  &  \\
0 & 0 & 0 & 0 & 0 & 1 & 5 & 10 & 10 & 5 & 1 &  &  &  &  &  &  &  &  \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 6 & 15 & 20 & 15 & 6 & 1 &  &  &  &  &  &  \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 7 & 21 & 35 & 35 & 21 & 7 & 1 &  &  &  &  \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 8 & 28 & 56 & 70 & 56 & 28 & 8 & 1 &  &  \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 9 & 36 & 84 & 126 & 126 & 84 & 36 & 9 & 1 \\
\end{array}\right] + m_{1,0}\left[\begin{array}{ccccccccccccccccccc}
0 &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
1 & 0 & 0 &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
0 & 1 & 1 & 0 & 0 &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
0 & 0 & 1 & 2 & 1 & 0 & 0 &  &  &  &  &  &  &  &  &  &  &  &  \\
0 & 0 & 0 & 1 & 3 & 3 & 1 & 0 & 0 &  &  &  &  &  &  &  &  &  &  \\
0 & 0 & 0 & 0 & 1 & 4 & 6 & 4 & 1 & 0 & 0 &  &  &  &  &  &  &  &  \\
0 & 0 & 0 & 0 & 0 & 1 & 5 & 10 & 10 & 5 & 1 & 0 & 0 &  &  &  &  &  &  \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 6 & 15 & 20 & 15 & 6 & 1 & 0 & 0 &  &  &  &  \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 7 & 21 & 35 & 35 & 21 & 7 & 1 & 0 & 0 &  &  \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 8 & 28 & 56 & 70 & 56 & 28 & 8 & 1 & 0 & 0 \\
\end{array}\right] +\\
& m_{2,0}\left[\begin{array}{ccccccccccccccccccc}
0 &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
0 & 0 & 0 &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
1 & 0 & 0 & 0 & 0 &  &  &  &  &  &  &  &  &  &  &  &  &  &  \\
1 & 1 & 1 & 0 & 0 & 0 & 0 &  &  &  &  &  &  &  &  &  &  &  &  \\
1 & 1 & 2 & 2 & 1 & 0 & 0 & 0 & 0 &  &  &  &  &  &  &  &  &  &  \\
1 & 1 & 2 & 3 & 4 & 3 & 1 & 0 & 0 & 0 & 0 &  &  &  &  &  &  &  &  \\
1 & 1 & 2 & 3 & 5 & 7 & 7 & 4 & 1 & 0 & 0 & 0 & 0 &  &  &  &  &  &  \\
1 & 1 & 2 & 3 & 5 & 8 & 12 & 14 & 11 & 5 & 1 & 0 & 0 & 0 & 0 &  &  &  &  \\
1 & 1 & 2 & 3 & 5 & 8 & 13 & 20 & 26 & 25 & 16 & 6 & 1 & 0 & 0 & 0 & 0 &  &  \\
1 & 1 & 2 & 3 & 5 & 8 & 13 & 21 & 33 & 46 & 51 & 41 & 22 & 7 & 1 & 0 & 0 & 0 & 0 \\
\end{array}\right]
\label{eq:stretched:pascal:like:three:splitted:matrix:expansion}
\end{split}
\end{equation}
\end{sidewaystable}

\subsection{Splitting renewal arrays with generalized $A$-sequence}

In this section we study the splitting of renewal arrays with a generalized
$A$-sequence:
\begin{displaymath}
    A(t) = \alpha + \beta\,t + \gamma\,t^{2}
\end{displaymath}
respect $2$ variables. In \autoref{eq:abstract:A:sequence:renewal:arrays:unfolded}
we report the unfold starting from row $1$; in \autoref{eq:abstract:A:sequence:renewal:arrays:splitted}
we report the splitting respect variables $m_{0,0}$ and $m_{1,0}$; finally,
in \autoref{eq:abstract:A:sequence:renewal:arrays:splitted:matrix:expansion} we
report the corresponding matrix expansion.

\begin{sidewaystable}
\scriptsize
\begin{equation}
m_{0,0}\left[\begin{array}{cccccccccc}
1 &  &  &  &  &  &  &  &  &  \\
\beta & \alpha &  &  &  &  &  &  &  &  \\
\alpha \gamma + \beta^{2} & 2 \alpha \beta & \alpha^{2} &  &  &  &  &  &  &  \\
3 \alpha \beta \gamma + \beta^{3} & 2 \alpha^{2} \gamma + 3 \alpha \beta^{2} & 3 \alpha^{2} \beta & \alpha^{3} &  &  &  &  &  &  \\
2 \alpha^{2} \gamma^{2} + 6 \alpha \beta^{2} \gamma + \beta^{4} & 8 \alpha^{2} \beta \gamma + 4 \alpha \beta^{3} & 3 \alpha^{3} \gamma + 6 \alpha^{2} \beta^{2} & 4 \alpha^{3} \beta & \alpha^{4} &  &  &  &  &  \\
10 \alpha^{2} \beta \gamma^{2} + 10 \alpha \beta^{3} \gamma + \beta^{5} & 5 \alpha^{3} \gamma^{2} + 20 \alpha^{2} \beta^{2} \gamma + 5 \alpha \beta^{4} & 15 \alpha^{3} \beta \gamma + 10 \alpha^{2} \beta^{3} & 4 \alpha^{4} \gamma + 10 \alpha^{3} \beta^{2} & 5 \alpha^{4} \beta & \alpha^{5} &  &  &  &  \\
5 \alpha^{3} \gamma^{3} + 30 \alpha^{2} \beta^{2} \gamma^{2} + 15 \alpha \beta^{4} \gamma + \beta^{6} & 30 \alpha^{3} \beta \gamma^{2} + 40 \alpha^{2} \beta^{3} \gamma + 6 \alpha \beta^{5} & 9 \alpha^{4} \gamma^{2} + 45 \alpha^{3} \beta^{2} \gamma + 15 \alpha^{2} \beta^{4} & 24 \alpha^{4} \beta \gamma + 20 \alpha^{3} \beta^{3} & 5 \alpha^{5} \gamma + 15 \alpha^{4} \beta^{2} & 6 \alpha^{5} \beta & \alpha^{6} &  &  &  \\
35 \alpha^{3} \beta \gamma^{3} + 70 \alpha^{2} \beta^{3} \gamma^{2} + 21 \alpha \beta^{5} \gamma + \beta^{7} & 14 \alpha^{4} \gamma^{3} + 105 \alpha^{3} \beta^{2} \gamma^{2} + 70 \alpha^{2} \beta^{4} \gamma + 7 \alpha \beta^{6} & 63 \alpha^{4} \beta \gamma^{2} + 105 \alpha^{3} \beta^{3} \gamma + 21 \alpha^{2} \beta^{5} & 14 \alpha^{5} \gamma^{2} + 84 \alpha^{4} \beta^{2} \gamma + 35 \alpha^{3} \beta^{4} & 35 \alpha^{5} \beta \gamma + 35 \alpha^{4} \beta^{3} & 6 \alpha^{6} \gamma + 21 \alpha^{5} \beta^{2} & 7 \alpha^{6} \beta & \alpha^{7} &  &  \\
14 \alpha^{4} \gamma^{4} + 140 \alpha^{3} \beta^{2} \gamma^{3} + 140 \alpha^{2} \beta^{4} \gamma^{2} + 28 \alpha \beta^{6} \gamma + \beta^{8} & 112 \alpha^{4} \beta \gamma^{3} + 280 \alpha^{3} \beta^{3} \gamma^{2} + 112 \alpha^{2} \beta^{5} \gamma + 8 \alpha \beta^{7} & 28 \alpha^{5} \gamma^{3} + 252 \alpha^{4} \beta^{2} \gamma^{2} + 210 \alpha^{3} \beta^{4} \gamma + 28 \alpha^{2} \beta^{6} & 112 \alpha^{5} \beta \gamma^{2} + 224 \alpha^{4} \beta^{3} \gamma + 56 \alpha^{3} \beta^{5} & 20 \alpha^{6} \gamma^{2} + 140 \alpha^{5} \beta^{2} \gamma + 70 \alpha^{4} \beta^{4} & 48 \alpha^{6} \beta \gamma + 56 \alpha^{5} \beta^{3} & 7 \alpha^{7} \gamma + 28 \alpha^{6} \beta^{2} & 8 \alpha^{7} \beta & \alpha^{8} &  \\
126 \alpha^{4} \beta \gamma^{4} + 420 \alpha^{3} \beta^{3} \gamma^{3} + 252 \alpha^{2} \beta^{5} \gamma^{2} + 36 \alpha \beta^{7} \gamma + \beta^{9} & 42 \alpha^{5} \gamma^{4} + 504 \alpha^{4} \beta^{2} \gamma^{3} + 630 \alpha^{3} \beta^{4} \gamma^{2} + 168 \alpha^{2} \beta^{6} \gamma + 9 \alpha \beta^{8} & 252 \alpha^{5} \beta \gamma^{3} + 756 \alpha^{4} \beta^{3} \gamma^{2} + 378 \alpha^{3} \beta^{5} \gamma + 36 \alpha^{2} \beta^{7} & 48 \alpha^{6} \gamma^{3} + 504 \alpha^{5} \beta^{2} \gamma^{2} + 504 \alpha^{4} \beta^{4} \gamma + 84 \alpha^{3} \beta^{6} & 180 \alpha^{6} \beta \gamma^{2} + 420 \alpha^{5} \beta^{3} \gamma + 126 \alpha^{4} \beta^{5} & 27 \alpha^{7} \gamma^{2} + 216 \alpha^{6} \beta^{2} \gamma + 126 \alpha^{5} \beta^{4} & 63 \alpha^{7} \beta \gamma + 84 \alpha^{6} \beta^{3} & 8 \alpha^{8} \gamma + 36 \alpha^{7} \beta^{2} & 9 \alpha^{8} \beta & \alpha^{9} \\
\end{array}\right]
\label{eq:abstract:A:sequence:renewal:arrays:unfolded}
\end{equation}
\begin{equation}
\left[\begin{array}{cccccccccc}
\textcolor{blue}{m_{0,0}} &  &  &  &  &  &  &  &  &  \\
\textcolor{blue}{m_{1,0}} & \alpha m_{0,0} &  &  &  &  &  &  &  &  \\
\alpha \gamma m_{0,0} + \beta m_{1,0} & \alpha \beta m_{0,0} + \alpha m_{1,0} & \alpha^{2} m_{0,0} &  &  &  &  &  &  &  \\
2 \alpha \beta \gamma m_{0,0} + \left(\alpha \gamma + \beta^{2}\right) m_{1,0} & 2 \alpha \beta m_{1,0} + \left(2 \alpha^{2} \gamma + \alpha \beta^{2}\right) m_{0,0} & 2 \alpha^{2} \beta m_{0,0} + \alpha^{2} m_{1,0} & \alpha^{3} m_{0,0} &  &  &  &  &  &  \\
\left(2 \alpha^{2} \gamma^{2} + 3 \alpha \beta^{2} \gamma\right) m_{0,0} + \left(3 \alpha \beta \gamma + \beta^{3}\right) m_{1,0} & \left(2 \alpha^{2} \gamma + 3 \alpha \beta^{2}\right) m_{1,0} + \left(6 \alpha^{2} \beta \gamma + \alpha \beta^{3}\right) m_{0,0} & 3 \alpha^{2} \beta m_{1,0} + \left(3 \alpha^{3} \gamma + 3 \alpha^{2} \beta^{2}\right) m_{0,0} & 3 \alpha^{3} \beta m_{0,0} + \alpha^{3} m_{1,0} & \alpha^{4} m_{0,0} &  &  &  &  &  \\
\left(8 \alpha^{2} \beta \gamma^{2} + 4 \alpha \beta^{3} \gamma\right) m_{0,0} + \left(2 \alpha^{2} \gamma^{2} + 6 \alpha \beta^{2} \gamma + \beta^{4}\right) m_{1,0} & \left(8 \alpha^{2} \beta \gamma + 4 \alpha \beta^{3}\right) m_{1,0} + \left(5 \alpha^{3} \gamma^{2} + 12 \alpha^{2} \beta^{2} \gamma + \alpha \beta^{4}\right) m_{0,0} & \left(3 \alpha^{3} \gamma + 6 \alpha^{2} \beta^{2}\right) m_{1,0} + \left(12 \alpha^{3} \beta \gamma + 4 \alpha^{2} \beta^{3}\right) m_{0,0} & 4 \alpha^{3} \beta m_{1,0} + \left(4 \alpha^{4} \gamma + 6 \alpha^{3} \beta^{2}\right) m_{0,0} & 4 \alpha^{4} \beta m_{0,0} + \alpha^{4} m_{1,0} & \alpha^{5} m_{0,0} &  &  &  &  \\
\left(5 \alpha^{3} \gamma^{3} + 20 \alpha^{2} \beta^{2} \gamma^{2} + 5 \alpha \beta^{4} \gamma\right) m_{0,0} + \left(10 \alpha^{2} \beta \gamma^{2} + 10 \alpha \beta^{3} \gamma + \beta^{5}\right) m_{1,0} & \left(5 \alpha^{3} \gamma^{2} + 20 \alpha^{2} \beta^{2} \gamma + 5 \alpha \beta^{4}\right) m_{1,0} + \left(25 \alpha^{3} \beta \gamma^{2} + 20 \alpha^{2} \beta^{3} \gamma + \alpha \beta^{5}\right) m_{0,0} & \left(15 \alpha^{3} \beta \gamma + 10 \alpha^{2} \beta^{3}\right) m_{1,0} + \left(9 \alpha^{4} \gamma^{2} + 30 \alpha^{3} \beta^{2} \gamma + 5 \alpha^{2} \beta^{4}\right) m_{0,0} & \left(4 \alpha^{4} \gamma + 10 \alpha^{3} \beta^{2}\right) m_{1,0} + \left(20 \alpha^{4} \beta \gamma + 10 \alpha^{3} \beta^{3}\right) m_{0,0} & 5 \alpha^{4} \beta m_{1,0} + \left(5 \alpha^{5} \gamma + 10 \alpha^{4} \beta^{2}\right) m_{0,0} & 5 \alpha^{5} \beta m_{0,0} + \alpha^{5} m_{1,0} & \alpha^{6} m_{0,0} &  &  &  \\
\left(30 \alpha^{3} \beta \gamma^{3} + 40 \alpha^{2} \beta^{3} \gamma^{2} + 6 \alpha \beta^{5} \gamma\right) m_{0,0} + \left(5 \alpha^{3} \gamma^{3} + 30 \alpha^{2} \beta^{2} \gamma^{2} + 15 \alpha \beta^{4} \gamma + \beta^{6}\right) m_{1,0} & \left(30 \alpha^{3} \beta \gamma^{2} + 40 \alpha^{2} \beta^{3} \gamma + 6 \alpha \beta^{5}\right) m_{1,0} + \left(14 \alpha^{4} \gamma^{3} + 75 \alpha^{3} \beta^{2} \gamma^{2} + 30 \alpha^{2} \beta^{4} \gamma + \alpha \beta^{6}\right) m_{0,0} & \left(9 \alpha^{4} \gamma^{2} + 45 \alpha^{3} \beta^{2} \gamma + 15 \alpha^{2} \beta^{4}\right) m_{1,0} + \left(54 \alpha^{4} \beta \gamma^{2} + 60 \alpha^{3} \beta^{3} \gamma + 6 \alpha^{2} \beta^{5}\right) m_{0,0} & \left(24 \alpha^{4} \beta \gamma + 20 \alpha^{3} \beta^{3}\right) m_{1,0} + \left(14 \alpha^{5} \gamma^{2} + 60 \alpha^{4} \beta^{2} \gamma + 15 \alpha^{3} \beta^{4}\right) m_{0,0} & \left(5 \alpha^{5} \gamma + 15 \alpha^{4} \beta^{2}\right) m_{1,0} + \left(30 \alpha^{5} \beta \gamma + 20 \alpha^{4} \beta^{3}\right) m_{0,0} & 6 \alpha^{5} \beta m_{1,0} + \left(6 \alpha^{6} \gamma + 15 \alpha^{5} \beta^{2}\right) m_{0,0} & 6 \alpha^{6} \beta m_{0,0} + \alpha^{6} m_{1,0} & \alpha^{7} m_{0,0} &  &  \\
\left(14 \alpha^{4} \gamma^{4} + 105 \alpha^{3} \beta^{2} \gamma^{3} + 70 \alpha^{2} \beta^{4} \gamma^{2} + 7 \alpha \beta^{6} \gamma\right) m_{0,0} + \left(35 \alpha^{3} \beta \gamma^{3} + 70 \alpha^{2} \beta^{3} \gamma^{2} + 21 \alpha \beta^{5} \gamma + \beta^{7}\right) m_{1,0} & \left(14 \alpha^{4} \gamma^{3} + 105 \alpha^{3} \beta^{2} \gamma^{2} + 70 \alpha^{2} \beta^{4} \gamma + 7 \alpha \beta^{6}\right) m_{1,0} + \left(98 \alpha^{4} \beta \gamma^{3} + 175 \alpha^{3} \beta^{3} \gamma^{2} + 42 \alpha^{2} \beta^{5} \gamma + \alpha \beta^{7}\right) m_{0,0} & \left(63 \alpha^{4} \beta \gamma^{2} + 105 \alpha^{3} \beta^{3} \gamma + 21 \alpha^{2} \beta^{5}\right) m_{1,0} + \left(28 \alpha^{5} \gamma^{3} + 189 \alpha^{4} \beta^{2} \gamma^{2} + 105 \alpha^{3} \beta^{4} \gamma + 7 \alpha^{2} \beta^{6}\right) m_{0,0} & \left(14 \alpha^{5} \gamma^{2} + 84 \alpha^{4} \beta^{2} \gamma + 35 \alpha^{3} \beta^{4}\right) m_{1,0} + \left(98 \alpha^{5} \beta \gamma^{2} + 140 \alpha^{4} \beta^{3} \gamma + 21 \alpha^{3} \beta^{5}\right) m_{0,0} & \left(35 \alpha^{5} \beta \gamma + 35 \alpha^{4} \beta^{3}\right) m_{1,0} + \left(20 \alpha^{6} \gamma^{2} + 105 \alpha^{5} \beta^{2} \gamma + 35 \alpha^{4} \beta^{4}\right) m_{0,0} & \left(6 \alpha^{6} \gamma + 21 \alpha^{5} \beta^{2}\right) m_{1,0} + \left(42 \alpha^{6} \beta \gamma + 35 \alpha^{5} \beta^{3}\right) m_{0,0} & 7 \alpha^{6} \beta m_{1,0} + \left(7 \alpha^{7} \gamma + 21 \alpha^{6} \beta^{2}\right) m_{0,0} & 7 \alpha^{7} \beta m_{0,0} + \alpha^{7} m_{1,0} & \alpha^{8} m_{0,0} &  \\
\left(112 \alpha^{4} \beta \gamma^{4} + 280 \alpha^{3} \beta^{3} \gamma^{3} + 112 \alpha^{2} \beta^{5} \gamma^{2} + 8 \alpha \beta^{7} \gamma\right) m_{0,0} + \left(14 \alpha^{4} \gamma^{4} + 140 \alpha^{3} \beta^{2} \gamma^{3} + 140 \alpha^{2} \beta^{4} \gamma^{2} + 28 \alpha \beta^{6} \gamma + \beta^{8}\right) m_{1,0} & \left(112 \alpha^{4} \beta \gamma^{3} + 280 \alpha^{3} \beta^{3} \gamma^{2} + 112 \alpha^{2} \beta^{5} \gamma + 8 \alpha \beta^{7}\right) m_{1,0} + \left(42 \alpha^{5} \gamma^{4} + 392 \alpha^{4} \beta^{2} \gamma^{3} + 350 \alpha^{3} \beta^{4} \gamma^{2} + 56 \alpha^{2} \beta^{6} \gamma + \alpha \beta^{8}\right) m_{0,0} & \left(28 \alpha^{5} \gamma^{3} + 252 \alpha^{4} \beta^{2} \gamma^{2} + 210 \alpha^{3} \beta^{4} \gamma + 28 \alpha^{2} \beta^{6}\right) m_{1,0} + \left(224 \alpha^{5} \beta \gamma^{3} + 504 \alpha^{4} \beta^{3} \gamma^{2} + 168 \alpha^{3} \beta^{5} \gamma + 8 \alpha^{2} \beta^{7}\right) m_{0,0} & \left(112 \alpha^{5} \beta \gamma^{2} + 224 \alpha^{4} \beta^{3} \gamma + 56 \alpha^{3} \beta^{5}\right) m_{1,0} + \left(48 \alpha^{6} \gamma^{3} + 392 \alpha^{5} \beta^{2} \gamma^{2} + 280 \alpha^{4} \beta^{4} \gamma + 28 \alpha^{3} \beta^{6}\right) m_{0,0} & \left(20 \alpha^{6} \gamma^{2} + 140 \alpha^{5} \beta^{2} \gamma + 70 \alpha^{4} \beta^{4}\right) m_{1,0} + \left(160 \alpha^{6} \beta \gamma^{2} + 280 \alpha^{5} \beta^{3} \gamma + 56 \alpha^{4} \beta^{5}\right) m_{0,0} & \left(48 \alpha^{6} \beta \gamma + 56 \alpha^{5} \beta^{3}\right) m_{1,0} + \left(27 \alpha^{7} \gamma^{2} + 168 \alpha^{6} \beta^{2} \gamma + 70 \alpha^{5} \beta^{4}\right) m_{0,0} & \left(7 \alpha^{7} \gamma + 28 \alpha^{6} \beta^{2}\right) m_{1,0} + \left(56 \alpha^{7} \beta \gamma + 56 \alpha^{6} \beta^{3}\right) m_{0,0} & 8 \alpha^{7} \beta m_{1,0} + \left(8 \alpha^{8} \gamma + 28 \alpha^{7} \beta^{2}\right) m_{0,0} & 8 \alpha^{8} \beta m_{0,0} + \alpha^{8} m_{1,0} & \alpha^{9} m_{0,0} \\
\end{array}\right]
\label{eq:abstract:A:sequence:renewal:arrays:splitted}
%\label{eq:generic:motzkin:Arec}
\end{equation}
\begin{equation}
\begin{split}
& m_{0,0}\left[\begin{array}{cccccccccc}
1 &  &  &  &  &  &  &  &  &  \\
0 & \alpha &  &  &  &  &  &  &  &  \\
\alpha \gamma & \alpha \beta & \alpha^{2} &  &  &  &  &  &  &  \\
2 \alpha \beta \gamma & 2 \alpha^{2} \gamma + \alpha \beta^{2} & 2 \alpha^{2} \beta & \alpha^{3} &  &  &  &  &  &  \\
2 \alpha^{2} \gamma^{2} + 3 \alpha \beta^{2} \gamma & 6 \alpha^{2} \beta \gamma + \alpha \beta^{3} & 3 \alpha^{3} \gamma + 3 \alpha^{2} \beta^{2} & 3 \alpha^{3} \beta & \alpha^{4} &  &  &  &  &  \\
8 \alpha^{2} \beta \gamma^{2} + 4 \alpha \beta^{3} \gamma & 5 \alpha^{3} \gamma^{2} + 12 \alpha^{2} \beta^{2} \gamma + \alpha \beta^{4} & 12 \alpha^{3} \beta \gamma + 4 \alpha^{2} \beta^{3} & 4 \alpha^{4} \gamma + 6 \alpha^{3} \beta^{2} & 4 \alpha^{4} \beta & \alpha^{5} &  &  &  &  \\
5 \alpha^{3} \gamma^{3} + 20 \alpha^{2} \beta^{2} \gamma^{2} + 5 \alpha \beta^{4} \gamma & 25 \alpha^{3} \beta \gamma^{2} + 20 \alpha^{2} \beta^{3} \gamma + \alpha \beta^{5} & 9 \alpha^{4} \gamma^{2} + 30 \alpha^{3} \beta^{2} \gamma + 5 \alpha^{2} \beta^{4} & 20 \alpha^{4} \beta \gamma + 10 \alpha^{3} \beta^{3} & 5 \alpha^{5} \gamma + 10 \alpha^{4} \beta^{2} & 5 \alpha^{5} \beta & \alpha^{6} &  &  &  \\
30 \alpha^{3} \beta \gamma^{3} + 40 \alpha^{2} \beta^{3} \gamma^{2} + 6 \alpha \beta^{5} \gamma & 14 \alpha^{4} \gamma^{3} + 75 \alpha^{3} \beta^{2} \gamma^{2} + 30 \alpha^{2} \beta^{4} \gamma + \alpha \beta^{6} & 54 \alpha^{4} \beta \gamma^{2} + 60 \alpha^{3} \beta^{3} \gamma + 6 \alpha^{2} \beta^{5} & 14 \alpha^{5} \gamma^{2} + 60 \alpha^{4} \beta^{2} \gamma + 15 \alpha^{3} \beta^{4} & 30 \alpha^{5} \beta \gamma + 20 \alpha^{4} \beta^{3} & 6 \alpha^{6} \gamma + 15 \alpha^{5} \beta^{2} & 6 \alpha^{6} \beta & \alpha^{7} &  &  \\
14 \alpha^{4} \gamma^{4} + 105 \alpha^{3} \beta^{2} \gamma^{3} + 70 \alpha^{2} \beta^{4} \gamma^{2} + 7 \alpha \beta^{6} \gamma & 98 \alpha^{4} \beta \gamma^{3} + 175 \alpha^{3} \beta^{3} \gamma^{2} + 42 \alpha^{2} \beta^{5} \gamma + \alpha \beta^{7} & 28 \alpha^{5} \gamma^{3} + 189 \alpha^{4} \beta^{2} \gamma^{2} + 105 \alpha^{3} \beta^{4} \gamma + 7 \alpha^{2} \beta^{6} & 98 \alpha^{5} \beta \gamma^{2} + 140 \alpha^{4} \beta^{3} \gamma + 21 \alpha^{3} \beta^{5} & 20 \alpha^{6} \gamma^{2} + 105 \alpha^{5} \beta^{2} \gamma + 35 \alpha^{4} \beta^{4} & 42 \alpha^{6} \beta \gamma + 35 \alpha^{5} \beta^{3} & 7 \alpha^{7} \gamma + 21 \alpha^{6} \beta^{2} & 7 \alpha^{7} \beta & \alpha^{8} &  \\
112 \alpha^{4} \beta \gamma^{4} + 280 \alpha^{3} \beta^{3} \gamma^{3} + 112 \alpha^{2} \beta^{5} \gamma^{2} + 8 \alpha \beta^{7} \gamma & 42 \alpha^{5} \gamma^{4} + 392 \alpha^{4} \beta^{2} \gamma^{3} + 350 \alpha^{3} \beta^{4} \gamma^{2} + 56 \alpha^{2} \beta^{6} \gamma + \alpha \beta^{8} & 224 \alpha^{5} \beta \gamma^{3} + 504 \alpha^{4} \beta^{3} \gamma^{2} + 168 \alpha^{3} \beta^{5} \gamma + 8 \alpha^{2} \beta^{7} & 48 \alpha^{6} \gamma^{3} + 392 \alpha^{5} \beta^{2} \gamma^{2} + 280 \alpha^{4} \beta^{4} \gamma + 28 \alpha^{3} \beta^{6} & 160 \alpha^{6} \beta \gamma^{2} + 280 \alpha^{5} \beta^{3} \gamma + 56 \alpha^{4} \beta^{5} & 27 \alpha^{7} \gamma^{2} + 168 \alpha^{6} \beta^{2} \gamma + 70 \alpha^{5} \beta^{4} & 56 \alpha^{7} \beta \gamma + 56 \alpha^{6} \beta^{3} & 8 \alpha^{8} \gamma + 28 \alpha^{7} \beta^{2} & 8 \alpha^{8} \beta & \alpha^{9} \\
\end{array}\right] \\
& + m_{1,0}\left[\begin{array}{cccccccccc}
0 &  &  &  &  &  &  &  &  &  \\
1 & 0 &  &  &  &  &  &  &  &  \\
\beta & \alpha & 0 &  &  &  &  &  &  &  \\
\alpha \gamma + \beta^{2} & 2 \alpha \beta & \alpha^{2} & 0 &  &  &  &  &  &  \\
3 \alpha \beta \gamma + \beta^{3} & 2 \alpha^{2} \gamma + 3 \alpha \beta^{2} & 3 \alpha^{2} \beta & \alpha^{3} & 0 &  &  &  &  &  \\
2 \alpha^{2} \gamma^{2} + 6 \alpha \beta^{2} \gamma + \beta^{4} & 8 \alpha^{2} \beta \gamma + 4 \alpha \beta^{3} & 3 \alpha^{3} \gamma + 6 \alpha^{2} \beta^{2} & 4 \alpha^{3} \beta & \alpha^{4} & 0 &  &  &  &  \\
10 \alpha^{2} \beta \gamma^{2} + 10 \alpha \beta^{3} \gamma + \beta^{5} & 5 \alpha^{3} \gamma^{2} + 20 \alpha^{2} \beta^{2} \gamma + 5 \alpha \beta^{4} & 15 \alpha^{3} \beta \gamma + 10 \alpha^{2} \beta^{3} & 4 \alpha^{4} \gamma + 10 \alpha^{3} \beta^{2} & 5 \alpha^{4} \beta & \alpha^{5} & 0 &  &  &  \\
5 \alpha^{3} \gamma^{3} + 30 \alpha^{2} \beta^{2} \gamma^{2} + 15 \alpha \beta^{4} \gamma + \beta^{6} & 30 \alpha^{3} \beta \gamma^{2} + 40 \alpha^{2} \beta^{3} \gamma + 6 \alpha \beta^{5} & 9 \alpha^{4} \gamma^{2} + 45 \alpha^{3} \beta^{2} \gamma + 15 \alpha^{2} \beta^{4} & 24 \alpha^{4} \beta \gamma + 20 \alpha^{3} \beta^{3} & 5 \alpha^{5} \gamma + 15 \alpha^{4} \beta^{2} & 6 \alpha^{5} \beta & \alpha^{6} & 0 &  &  \\
35 \alpha^{3} \beta \gamma^{3} + 70 \alpha^{2} \beta^{3} \gamma^{2} + 21 \alpha \beta^{5} \gamma + \beta^{7} & 14 \alpha^{4} \gamma^{3} + 105 \alpha^{3} \beta^{2} \gamma^{2} + 70 \alpha^{2} \beta^{4} \gamma + 7 \alpha \beta^{6} & 63 \alpha^{4} \beta \gamma^{2} + 105 \alpha^{3} \beta^{3} \gamma + 21 \alpha^{2} \beta^{5} & 14 \alpha^{5} \gamma^{2} + 84 \alpha^{4} \beta^{2} \gamma + 35 \alpha^{3} \beta^{4} & 35 \alpha^{5} \beta \gamma + 35 \alpha^{4} \beta^{3} & 6 \alpha^{6} \gamma + 21 \alpha^{5} \beta^{2} & 7 \alpha^{6} \beta & \alpha^{7} & 0 &  \\
14 \alpha^{4} \gamma^{4} + 140 \alpha^{3} \beta^{2} \gamma^{3} + 140 \alpha^{2} \beta^{4} \gamma^{2} + 28 \alpha \beta^{6} \gamma + \beta^{8} & 112 \alpha^{4} \beta \gamma^{3} + 280 \alpha^{3} \beta^{3} \gamma^{2} + 112 \alpha^{2} \beta^{5} \gamma + 8 \alpha \beta^{7} & 28 \alpha^{5} \gamma^{3} + 252 \alpha^{4} \beta^{2} \gamma^{2} + 210 \alpha^{3} \beta^{4} \gamma + 28 \alpha^{2} \beta^{6} & 112 \alpha^{5} \beta \gamma^{2} + 224 \alpha^{4} \beta^{3} \gamma + 56 \alpha^{3} \beta^{5} & 20 \alpha^{6} \gamma^{2} + 140 \alpha^{5} \beta^{2} \gamma + 70 \alpha^{4} \beta^{4} & 48 \alpha^{6} \beta \gamma + 56 \alpha^{5} \beta^{3} & 7 \alpha^{7} \gamma + 28 \alpha^{6} \beta^{2} & 8 \alpha^{7} \beta & \alpha^{8} & 0 \\
\end{array}\right]
\end{split}
\label{eq:abstract:A:sequence:renewal:arrays:splitted:matrix:expansion}
\end{equation}
\end{sidewaystable}

\subsection{Complete matrix expansions of $\mathcal{C}$}

All the content above let us to observe that after the first unfolding
macro-step, where elements in a purely symbolic matrix are unfolded
according $A$-sequences and $Z$-sequences, starting from a given row $r$,
then in the first $r$ rows there are actually ${{r}\choose{2}}$ free variables.
What we've done so far is to fix $r$ variables lying on the top segment of
column $0$, namely variables $c_{0,0}\ldots c_{r-1,0}$, using $c$ as
generic symbol for the matrix.

Therefore we've:
\begin{displaymath}
    {{{{r}\choose{2}}}\choose{r}}
\end{displaymath}
possible choices of $r$ variables in order to build a matrix expansion using $r$ matrices.
In the following three sections we report \emph{complete} matrix expansions of $\mathcal{C}$
where variables rest on the top segment of column $0$, on the main diagonal and, finally,
on the last but one row of the unfolding, respectively.

\subsubsection{Free variables lying on column $0$}

In \autoref{eq:complete:splitting:catalan:variables:on:column:zero},
\autoref{eq:complete:expansion:catalan:variables:on:column:zero} and
\autoref{eq:complete:dependencies:catalan:variables:on:column:zero}
we report splitting, matrix expansion and dependencies of $\mathcal{C}$
using variables $c_{0,0},c_{1,0},\ldots, c_{9,0}$, respectively.

\begin{sidewaystable}
\scriptsize
\begin{equation}
\left[\begin{array}{cccccccccc}
\textcolor{blue}{c_{0,0}} &  &  &  &  &  &  &  &  &  \\
\textcolor{blue}{c_{1,0}} & c_{0,0} &  &  &  &  &  &  &  &  \\
\textcolor{blue}{c_{2,0}} & c_{0,0} + c_{1,0} & c_{0,0} &  &  &  &  &  &  &  \\
\textcolor{blue}{c_{3,0}} & 2 c_{0,0} + c_{1,0} + c_{2,0} & 2 c_{0,0} + c_{1,0} & c_{0,0} &  &  &  &  &  &  \\
\textcolor{blue}{c_{4,0}} & 5 c_{0,0} + 2 c_{1,0} + c_{2,0} + c_{3,0} & 5 c_{0,0} + 2 c_{1,0} + c_{2,0} & 3 c_{0,0} + c_{1,0} & c_{0,0} &  &  &  &  &  \\
\textcolor{blue}{c_{5,0}} & 14 c_{0,0} + 5 c_{1,0} + 2 c_{2,0} + c_{3,0} + c_{4,0} & 14 c_{0,0} + 5 c_{1,0} + 2 c_{2,0} + c_{3,0} & 9 c_{0,0} + 3 c_{1,0} + c_{2,0} & 4 c_{0,0} + c_{1,0} & c_{0,0} &  &  &  &  \\
\textcolor{blue}{c_{6,0}} & 42 c_{0,0} + 14 c_{1,0} + 5 c_{2,0} + 2 c_{3,0} + c_{4,0} + c_{5,0} & 42 c_{0,0} + 14 c_{1,0} + 5 c_{2,0} + 2 c_{3,0} + c_{4,0} & 28 c_{0,0} + 9 c_{1,0} + 3 c_{2,0} + c_{3,0} & 14 c_{0,0} + 4 c_{1,0} + c_{2,0} & 5 c_{0,0} + c_{1,0} & c_{0,0} &  &  &  \\
\textcolor{blue}{c_{7,0}} & 132 c_{0,0} + 42 c_{1,0} + 14 c_{2,0} + 5 c_{3,0} + 2 c_{4,0} + c_{5,0} + c_{6,0} & 132 c_{0,0} + 42 c_{1,0} + 14 c_{2,0} + 5 c_{3,0} + 2 c_{4,0} + c_{5,0} & 90 c_{0,0} + 28 c_{1,0} + 9 c_{2,0} + 3 c_{3,0} + c_{4,0} & 48 c_{0,0} + 14 c_{1,0} + 4 c_{2,0} + c_{3,0} & 20 c_{0,0} + 5 c_{1,0} + c_{2,0} & 6 c_{0,0} + c_{1,0} & c_{0,0} &  &  \\
\textcolor{blue}{c_{8,0}} & 429 c_{0,0} + 132 c_{1,0} + 42 c_{2,0} + 14 c_{3,0} + 5 c_{4,0} + 2 c_{5,0} + c_{6,0} + c_{7,0} & 429 c_{0,0} + 132 c_{1,0} + 42 c_{2,0} + 14 c_{3,0} + 5 c_{4,0} + 2 c_{5,0} + c_{6,0} & 297 c_{0,0} + 90 c_{1,0} + 28 c_{2,0} + 9 c_{3,0} + 3 c_{4,0} + c_{5,0} & 165 c_{0,0} + 48 c_{1,0} + 14 c_{2,0} + 4 c_{3,0} + c_{4,0} & 75 c_{0,0} + 20 c_{1,0} + 5 c_{2,0} + c_{3,0} & 27 c_{0,0} + 6 c_{1,0} + c_{2,0} & 7 c_{0,0} + c_{1,0} & c_{0,0} &  \\
\textcolor{blue}{c_{9,0}} & 1430 c_{0,0} + 429 c_{1,0} + 132 c_{2,0} + 42 c_{3,0} + 14 c_{4,0} + 5 c_{5,0} + 2 c_{6,0} + c_{7,0} + c_{8,0} & 1430 c_{0,0} + 429 c_{1,0} + 132 c_{2,0} + 42 c_{3,0} + 14 c_{4,0} + 5 c_{5,0} + 2 c_{6,0} + c_{7,0} & 1001 c_{0,0} + 297 c_{1,0} + 90 c_{2,0} + 28 c_{3,0} + 9 c_{4,0} + 3 c_{5,0} + c_{6,0} & 572 c_{0,0} + 165 c_{1,0} + 48 c_{2,0} + 14 c_{3,0} + 4 c_{4,0} + c_{5,0} & 275 c_{0,0} + 75 c_{1,0} + 20 c_{2,0} + 5 c_{3,0} + c_{4,0} & 110 c_{0,0} + 27 c_{1,0} + 6 c_{2,0} + c_{3,0} & 35 c_{0,0} + 7 c_{1,0} + c_{2,0} & 8 c_{0,0} + c_{1,0} & c_{0,0} \\
\end{array}\right]
\label{eq:complete:splitting:catalan:variables:on:column:zero}
\end{equation}
\begin{equation}
    \begin{split}
    & c_{0,0}\left[\begin{array}{cccccccccc}
    1 &  &  &  &  &  &  &  &  &  \\
    0 & 1 &  &  &  &  &  &  &  &  \\
    0 & 1 & 1 &  &  &  &  &  &  &  \\
    0 & 2 & 2 & 1 &  &  &  &  &  &  \\
    0 & 5 & 5 & 3 & 1 &  &  &  &  &  \\
    0 & 14 & 14 & 9 & 4 & 1 &  &  &  &  \\
    0 & 42 & 42 & 28 & 14 & 5 & 1 &  &  &  \\
    0 & 132 & 132 & 90 & 48 & 20 & 6 & 1 &  &  \\
    0 & 429 & 429 & 297 & 165 & 75 & 27 & 7 & 1 &  \\
    0 & 1430 & 1430 & 1001 & 572 & 275 & 110 & 35 & 8 & 1 \\
    \end{array}\right] + c_{1,0}\left[\begin{array}{cccccccccc}
    0 &  &  &  &  &  &  &  &  &  \\
    1 & 0 &  &  &  &  &  &  &  &  \\
    0 & 1 & 0 &  &  &  &  &  &  &  \\
    0 & 1 & 1 & 0 &  &  &  &  &  &  \\
    0 & 2 & 2 & 1 & 0 &  &  &  &  &  \\
    0 & 5 & 5 & 3 & 1 & 0 &  &  &  &  \\
    0 & 14 & 14 & 9 & 4 & 1 & 0 &  &  &  \\
    0 & 42 & 42 & 28 & 14 & 5 & 1 & 0 &  &  \\
    0 & 132 & 132 & 90 & 48 & 20 & 6 & 1 & 0 &  \\
    0 & 429 & 429 & 297 & 165 & 75 & 27 & 7 & 1 & 0 \\
    \end{array}\right] + c_{2,0}\left[\begin{array}{cccccccccc}
    0 &  &  &  &  &  &  &  &  &  \\
    0 & 0 &  &  &  &  &  &  &  &  \\
    1 & 0 & 0 &  &  &  &  &  &  &  \\
    0 & 1 & 0 & 0 &  &  &  &  &  &  \\
    0 & 1 & 1 & 0 & 0 &  &  &  &  &  \\
    0 & 2 & 2 & 1 & 0 & 0 &  &  &  &  \\
    0 & 5 & 5 & 3 & 1 & 0 & 0 &  &  &  \\
    0 & 14 & 14 & 9 & 4 & 1 & 0 & 0 &  &  \\
    0 & 42 & 42 & 28 & 14 & 5 & 1 & 0 & 0 &  \\
    0 & 132 & 132 & 90 & 48 & 20 & 6 & 1 & 0 & 0 \\
    \end{array}\right]\\
    & + c_{3,0}\left[\begin{array}{cccccccccc}
    0 &  &  &  &  &  &  &  &  &  \\
    0 & 0 &  &  &  &  &  &  &  &  \\
    0 & 0 & 0 &  &  &  &  &  &  &  \\
    1 & 0 & 0 & 0 &  &  &  &  &  &  \\
    0 & 1 & 0 & 0 & 0 &  &  &  &  &  \\
    0 & 1 & 1 & 0 & 0 & 0 &  &  &  &  \\
    0 & 2 & 2 & 1 & 0 & 0 & 0 &  &  &  \\
    0 & 5 & 5 & 3 & 1 & 0 & 0 & 0 &  &  \\
    0 & 14 & 14 & 9 & 4 & 1 & 0 & 0 & 0 &  \\
    0 & 42 & 42 & 28 & 14 & 5 & 1 & 0 & 0 & 0 \\
    \end{array}\right] + c_{4,0}\left[\begin{array}{cccccccccc}
    0 &  &  &  &  &  &  &  &  &  \\
    0 & 0 &  &  &  &  &  &  &  &  \\
    0 & 0 & 0 &  &  &  &  &  &  &  \\
    0 & 0 & 0 & 0 &  &  &  &  &  &  \\
    1 & 0 & 0 & 0 & 0 &  &  &  &  &  \\
    0 & 1 & 0 & 0 & 0 & 0 &  &  &  &  \\
    0 & 1 & 1 & 0 & 0 & 0 & 0 &  &  &  \\
    0 & 2 & 2 & 1 & 0 & 0 & 0 & 0 &  &  \\
    0 & 5 & 5 & 3 & 1 & 0 & 0 & 0 & 0 &  \\
    0 & 14 & 14 & 9 & 4 & 1 & 0 & 0 & 0 & 0 \\
    \end{array}\right] + c_{5,0}\left[\begin{array}{cccccccccc}
    0 &  &  &  &  &  &  &  &  &  \\
    0 & 0 &  &  &  &  &  &  &  &  \\
    0 & 0 & 0 &  &  &  &  &  &  &  \\
    0 & 0 & 0 & 0 &  &  &  &  &  &  \\
    0 & 0 & 0 & 0 & 0 &  &  &  &  &  \\
    1 & 0 & 0 & 0 & 0 & 0 &  &  &  &  \\
    0 & 1 & 0 & 0 & 0 & 0 & 0 &  &  &  \\
    0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 &  &  \\
    0 & 2 & 2 & 1 & 0 & 0 & 0 & 0 & 0 &  \\
    0 & 5 & 5 & 3 & 1 & 0 & 0 & 0 & 0 & 0 \\
    \end{array}\right] + c_{6,0}\left[\begin{array}{cccccccccc}
    0 &  &  &  &  &  &  &  &  &  \\
    0 & 0 &  &  &  &  &  &  &  &  \\
    0 & 0 & 0 &  &  &  &  &  &  &  \\
    0 & 0 & 0 & 0 &  &  &  &  &  &  \\
    0 & 0 & 0 & 0 & 0 &  &  &  &  &  \\
    0 & 0 & 0 & 0 & 0 & 0 &  &  &  &  \\
    1 & 0 & 0 & 0 & 0 & 0 & 0 &  &  &  \\
    0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 &  &  \\
    0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 &  \\
    0 & 2 & 2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
    \end{array}\right]\\
    & + c_{7,0}\left[\begin{array}{cccccccccc}
    0 &  &  &  &  &  &  &  &  &  \\
    0 & 0 &  &  &  &  &  &  &  &  \\
    0 & 0 & 0 &  &  &  &  &  &  &  \\
    0 & 0 & 0 & 0 &  &  &  &  &  &  \\
    0 & 0 & 0 & 0 & 0 &  &  &  &  &  \\
    0 & 0 & 0 & 0 & 0 & 0 &  &  &  &  \\
    0 & 0 & 0 & 0 & 0 & 0 & 0 &  &  &  \\
    1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &  &  \\
    0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &  \\
    0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
    \end{array}\right] + c_{8,0}\left[\begin{array}{cccccccccc}
    0 &  &  &  &  &  &  &  &  &  \\
    0 & 0 &  &  &  &  &  &  &  &  \\
    0 & 0 & 0 &  &  &  &  &  &  &  \\
    0 & 0 & 0 & 0 &  &  &  &  &  &  \\
    0 & 0 & 0 & 0 & 0 &  &  &  &  &  \\
    0 & 0 & 0 & 0 & 0 & 0 &  &  &  &  \\
    0 & 0 & 0 & 0 & 0 & 0 & 0 &  &  &  \\
    0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &  &  \\
    1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &  \\
    0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
    \end{array}\right] + c_{9,0}\left[\begin{array}{cccccccccc}
    0 &  &  &  &  &  &  &  &  &  \\
    0 & 0 &  &  &  &  &  &  &  &  \\
    0 & 0 & 0 &  &  &  &  &  &  &  \\
    0 & 0 & 0 & 0 &  &  &  &  &  &  \\
    0 & 0 & 0 & 0 & 0 &  &  &  &  &  \\
    0 & 0 & 0 & 0 & 0 & 0 &  &  &  &  \\
    0 & 0 & 0 & 0 & 0 & 0 & 0 &  &  &  \\
    0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &  &  \\
    0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &  \\
    1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
    \end{array}\right]
    \end{split}
\label{eq:complete:expansion:catalan:variables:on:column:zero}
\end{equation}
\begin{equation}
\text{Buondary conditions on free variables:}\quad\left \{ c_{1,0} : c_{0,0}, \quad c_{2,0} : 2 c_{0,0}, \quad c_{3,0} : 5 c_{0,0}, \quad c_{4,0} : 14 c_{0,0}, \quad c_{5,0} : 42 c_{0,0}, \quad c_{6,0} : 132 c_{0,0}, \quad c_{7,0} : 429 c_{0,0}, \quad c_{8,0} : 1430 c_{0,0}, \quad c_{9,0} : 4862 c_{0,0}\right \}
\label{eq:complete:dependencies:catalan:variables:on:column:zero}
\end{equation}
\end{sidewaystable}

\subsubsection{Free variables lying on main diagonal}

In \autoref{eq:complete:splitting:catalan:variables:on:main:diagonal},
\autoref{eq:complete:expansion:catalan:variables:on:main:diagonal} and
\autoref{eq:complete:dependencies:catalan:variables:on:main:diagonal}
we report splitting, matrix expansion and dependencies of $\mathcal{C}$
using variables $c_{0,0},c_{1,1},\ldots, c_{9,9}$, respectively.

\begin{sidewaystable}
\scriptsize
\begin{equation}
\left[\begin{array}{cccccccccc}
\textcolor{blue}{c_{0,0}} &  &  &  &  &  &  &  &  &  \\
c_{0,0} & \textcolor{blue}{c_{1,1}} &  &  &  &  &  &  &  &  \\
c_{0,0} + c_{1,1} & c_{0,0} + c_{1,1} & \textcolor{blue}{c_{2,2}} &  &  &  &  &  &  &  \\
2 c_{0,0} + 2 c_{1,1} + c_{2,2} & 2 c_{0,0} + 2 c_{1,1} + c_{2,2} & c_{0,0} + c_{1,1} + c_{2,2} & \textcolor{blue}{c_{3,3}} &  &  &  &  &  &  \\
5 c_{0,0} + 5 c_{1,1} + 3 c_{2,2} + c_{3,3} & 5 c_{0,0} + 5 c_{1,1} + 3 c_{2,2} + c_{3,3} & 3 c_{0,0} + 3 c_{1,1} + 2 c_{2,2} + c_{3,3} & c_{0,0} + c_{1,1} + c_{2,2} + c_{3,3} & \textcolor{blue}{c_{4,4}} &  &  &  &  &  \\
14 c_{0,0} + 14 c_{1,1} + 9 c_{2,2} + 4 c_{3,3} + c_{4,4} & 14 c_{0,0} + 14 c_{1,1} + 9 c_{2,2} + 4 c_{3,3} + c_{4,4} & 9 c_{0,0} + 9 c_{1,1} + 6 c_{2,2} + 3 c_{3,3} + c_{4,4} & 4 c_{0,0} + 4 c_{1,1} + 3 c_{2,2} + 2 c_{3,3} + c_{4,4} & c_{0,0} + c_{1,1} + c_{2,2} + c_{3,3} + c_{4,4} & \textcolor{blue}{c_{5,5}} &  &  &  &  \\
42 c_{0,0} + 42 c_{1,1} + 28 c_{2,2} + 14 c_{3,3} + 5 c_{4,4} + c_{5,5} & 42 c_{0,0} + 42 c_{1,1} + 28 c_{2,2} + 14 c_{3,3} + 5 c_{4,4} + c_{5,5} & 28 c_{0,0} + 28 c_{1,1} + 19 c_{2,2} + 10 c_{3,3} + 4 c_{4,4} + c_{5,5} & 14 c_{0,0} + 14 c_{1,1} + 10 c_{2,2} + 6 c_{3,3} + 3 c_{4,4} + c_{5,5} & 5 c_{0,0} + 5 c_{1,1} + 4 c_{2,2} + 3 c_{3,3} + 2 c_{4,4} + c_{5,5} & c_{0,0} + c_{1,1} + c_{2,2} + c_{3,3} + c_{4,4} + c_{5,5} & \textcolor{blue}{c_{6,6}} &  &  &  \\
132 c_{0,0} + 132 c_{1,1} + 90 c_{2,2} + 48 c_{3,3} + 20 c_{4,4} + 6 c_{5,5} + c_{6,6} & 132 c_{0,0} + 132 c_{1,1} + 90 c_{2,2} + 48 c_{3,3} + 20 c_{4,4} + 6 c_{5,5} + c_{6,6} & 90 c_{0,0} + 90 c_{1,1} + 62 c_{2,2} + 34 c_{3,3} + 15 c_{4,4} + 5 c_{5,5} + c_{6,6} & 48 c_{0,0} + 48 c_{1,1} + 34 c_{2,2} + 20 c_{3,3} + 10 c_{4,4} + 4 c_{5,5} + c_{6,6} & 20 c_{0,0} + 20 c_{1,1} + 15 c_{2,2} + 10 c_{3,3} + 6 c_{4,4} + 3 c_{5,5} + c_{6,6} & 6 c_{0,0} + 6 c_{1,1} + 5 c_{2,2} + 4 c_{3,3} + 3 c_{4,4} + 2 c_{5,5} + c_{6,6} & c_{0,0} + c_{1,1} + c_{2,2} + c_{3,3} + c_{4,4} + c_{5,5} + c_{6,6} & \textcolor{blue}{c_{7,7}} &  &  \\
429 c_{0,0} + 429 c_{1,1} + 297 c_{2,2} + 165 c_{3,3} + 75 c_{4,4} + 27 c_{5,5} + 7 c_{6,6} + c_{7,7} & 429 c_{0,0} + 429 c_{1,1} + 297 c_{2,2} + 165 c_{3,3} + 75 c_{4,4} + 27 c_{5,5} + 7 c_{6,6} + c_{7,7} & 297 c_{0,0} + 297 c_{1,1} + 207 c_{2,2} + 117 c_{3,3} + 55 c_{4,4} + 21 c_{5,5} + 6 c_{6,6} + c_{7,7} & 165 c_{0,0} + 165 c_{1,1} + 117 c_{2,2} + 69 c_{3,3} + 35 c_{4,4} + 15 c_{5,5} + 5 c_{6,6} + c_{7,7} & 75 c_{0,0} + 75 c_{1,1} + 55 c_{2,2} + 35 c_{3,3} + 20 c_{4,4} + 10 c_{5,5} + 4 c_{6,6} + c_{7,7} & 27 c_{0,0} + 27 c_{1,1} + 21 c_{2,2} + 15 c_{3,3} + 10 c_{4,4} + 6 c_{5,5} + 3 c_{6,6} + c_{7,7} & 7 c_{0,0} + 7 c_{1,1} + 6 c_{2,2} + 5 c_{3,3} + 4 c_{4,4} + 3 c_{5,5} + 2 c_{6,6} + c_{7,7} & c_{0,0} + c_{1,1} + c_{2,2} + c_{3,3} + c_{4,4} + c_{5,5} + c_{6,6} + c_{7,7} & \textcolor{blue}{c_{8,8}} &  \\
1430 c_{0,0} + 1430 c_{1,1} + 1001 c_{2,2} + 572 c_{3,3} + 275 c_{4,4} + 110 c_{5,5} + 35 c_{6,6} + 8 c_{7,7} + c_{8,8} & 1430 c_{0,0} + 1430 c_{1,1} + 1001 c_{2,2} + 572 c_{3,3} + 275 c_{4,4} + 110 c_{5,5} + 35 c_{6,6} + 8 c_{7,7} + c_{8,8} & 1001 c_{0,0} + 1001 c_{1,1} + 704 c_{2,2} + 407 c_{3,3} + 200 c_{4,4} + 83 c_{5,5} + 28 c_{6,6} + 7 c_{7,7} + c_{8,8} & 572 c_{0,0} + 572 c_{1,1} + 407 c_{2,2} + 242 c_{3,3} + 125 c_{4,4} + 56 c_{5,5} + 21 c_{6,6} + 6 c_{7,7} + c_{8,8} & 275 c_{0,0} + 275 c_{1,1} + 200 c_{2,2} + 125 c_{3,3} + 70 c_{4,4} + 35 c_{5,5} + 15 c_{6,6} + 5 c_{7,7} + c_{8,8} & 110 c_{0,0} + 110 c_{1,1} + 83 c_{2,2} + 56 c_{3,3} + 35 c_{4,4} + 20 c_{5,5} + 10 c_{6,6} + 4 c_{7,7} + c_{8,8} & 35 c_{0,0} + 35 c_{1,1} + 28 c_{2,2} + 21 c_{3,3} + 15 c_{4,4} + 10 c_{5,5} + 6 c_{6,6} + 3 c_{7,7} + c_{8,8} & 8 c_{0,0} + 8 c_{1,1} + 7 c_{2,2} + 6 c_{3,3} + 5 c_{4,4} + 4 c_{5,5} + 3 c_{6,6} + 2 c_{7,7} + c_{8,8} & c_{0,0} + c_{1,1} + c_{2,2} + c_{3,3} + c_{4,4} + c_{5,5} + c_{6,6} + c_{7,7} + c_{8,8} & \textcolor{blue}{c_{9,9}} \\
\end{array}\right]
\label{eq:complete:splitting:catalan:variables:on:main:diagonal}
\end{equation}
\begin{equation}
    \begin{split}
& c_{0,0}\left[\begin{array}{cccccccccc}
1 &  &  &  &  &  &  &  &  &  \\
1 & 0 &  &  &  &  &  &  &  &  \\
1 & 1 & 0 &  &  &  &  &  &  &  \\
2 & 2 & 1 & 0 &  &  &  &  &  &  \\
5 & 5 & 3 & 1 & 0 &  &  &  &  &  \\
14 & 14 & 9 & 4 & 1 & 0 &  &  &  &  \\
42 & 42 & 28 & 14 & 5 & 1 & 0 &  &  &  \\
132 & 132 & 90 & 48 & 20 & 6 & 1 & 0 &  &  \\
429 & 429 & 297 & 165 & 75 & 27 & 7 & 1 & 0 &  \\
1430 & 1430 & 1001 & 572 & 275 & 110 & 35 & 8 & 1 & 0 \\
\end{array}\right] + c_{1,1}\left[\begin{array}{cccccccccc}
0 &  &  &  &  &  &  &  &  &  \\
0 & 1 &  &  &  &  &  &  &  &  \\
1 & 1 & 0 &  &  &  &  &  &  &  \\
2 & 2 & 1 & 0 &  &  &  &  &  &  \\
5 & 5 & 3 & 1 & 0 &  &  &  &  &  \\
14 & 14 & 9 & 4 & 1 & 0 &  &  &  &  \\
42 & 42 & 28 & 14 & 5 & 1 & 0 &  &  &  \\
132 & 132 & 90 & 48 & 20 & 6 & 1 & 0 &  &  \\
429 & 429 & 297 & 165 & 75 & 27 & 7 & 1 & 0 &  \\
1430 & 1430 & 1001 & 572 & 275 & 110 & 35 & 8 & 1 & 0 \\
\end{array}\right] + c_{2,2}\left[\begin{array}{cccccccccc}
0 &  &  &  &  &  &  &  &  &  \\
0 & 0 &  &  &  &  &  &  &  &  \\
0 & 0 & 1 &  &  &  &  &  &  &  \\
1 & 1 & 1 & 0 &  &  &  &  &  &  \\
3 & 3 & 2 & 1 & 0 &  &  &  &  &  \\
9 & 9 & 6 & 3 & 1 & 0 &  &  &  &  \\
28 & 28 & 19 & 10 & 4 & 1 & 0 &  &  &  \\
90 & 90 & 62 & 34 & 15 & 5 & 1 & 0 &  &  \\
297 & 297 & 207 & 117 & 55 & 21 & 6 & 1 & 0 &  \\
1001 & 1001 & 704 & 407 & 200 & 83 & 28 & 7 & 1 & 0 \\
\end{array}\right]\\
& + c_{3,3}\left[\begin{array}{cccccccccc}
0 &  &  &  &  &  &  &  &  &  \\
0 & 0 &  &  &  &  &  &  &  &  \\
0 & 0 & 0 &  &  &  &  &  &  &  \\
0 & 0 & 0 & 1 &  &  &  &  &  &  \\
1 & 1 & 1 & 1 & 0 &  &  &  &  &  \\
4 & 4 & 3 & 2 & 1 & 0 &  &  &  &  \\
14 & 14 & 10 & 6 & 3 & 1 & 0 &  &  &  \\
48 & 48 & 34 & 20 & 10 & 4 & 1 & 0 &  &  \\
165 & 165 & 117 & 69 & 35 & 15 & 5 & 1 & 0 &  \\
572 & 572 & 407 & 242 & 125 & 56 & 21 & 6 & 1 & 0 \\
\end{array}\right] + c_{4,4}\left[\begin{array}{cccccccccc}
0 &  &  &  &  &  &  &  &  &  \\
0 & 0 &  &  &  &  &  &  &  &  \\
0 & 0 & 0 &  &  &  &  &  &  &  \\
0 & 0 & 0 & 0 &  &  &  &  &  &  \\
0 & 0 & 0 & 0 & 1 &  &  &  &  &  \\
1 & 1 & 1 & 1 & 1 & 0 &  &  &  &  \\
5 & 5 & 4 & 3 & 2 & 1 & 0 &  &  &  \\
20 & 20 & 15 & 10 & 6 & 3 & 1 & 0 &  &  \\
75 & 75 & 55 & 35 & 20 & 10 & 4 & 1 & 0 &  \\
275 & 275 & 200 & 125 & 70 & 35 & 15 & 5 & 1 & 0 \\
\end{array}\right] + c_{5,5}\left[\begin{array}{cccccccccc}
0 &  &  &  &  &  &  &  &  &  \\
0 & 0 &  &  &  &  &  &  &  &  \\
0 & 0 & 0 &  &  &  &  &  &  &  \\
0 & 0 & 0 & 0 &  &  &  &  &  &  \\
0 & 0 & 0 & 0 & 0 &  &  &  &  &  \\
0 & 0 & 0 & 0 & 0 & 1 &  &  &  &  \\
1 & 1 & 1 & 1 & 1 & 1 & 0 &  &  &  \\
6 & 6 & 5 & 4 & 3 & 2 & 1 & 0 &  &  \\
27 & 27 & 21 & 15 & 10 & 6 & 3 & 1 & 0 &  \\
110 & 110 & 83 & 56 & 35 & 20 & 10 & 4 & 1 & 0 \\
\end{array}\right]\\
& + c_{6,6}\left[\begin{array}{cccccccccc}
0 &  &  &  &  &  &  &  &  &  \\
0 & 0 &  &  &  &  &  &  &  &  \\
0 & 0 & 0 &  &  &  &  &  &  &  \\
0 & 0 & 0 & 0 &  &  &  &  &  &  \\
0 & 0 & 0 & 0 & 0 &  &  &  &  &  \\
0 & 0 & 0 & 0 & 0 & 0 &  &  &  &  \\
0 & 0 & 0 & 0 & 0 & 0 & 1 &  &  &  \\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 &  &  \\
7 & 7 & 6 & 5 & 4 & 3 & 2 & 1 & 0 &  \\
35 & 35 & 28 & 21 & 15 & 10 & 6 & 3 & 1 & 0 \\
\end{array}\right] + c_{7,7}\left[\begin{array}{cccccccccc}
0 &  &  &  &  &  &  &  &  &  \\
0 & 0 &  &  &  &  &  &  &  &  \\
0 & 0 & 0 &  &  &  &  &  &  &  \\
0 & 0 & 0 & 0 &  &  &  &  &  &  \\
0 & 0 & 0 & 0 & 0 &  &  &  &  &  \\
0 & 0 & 0 & 0 & 0 & 0 &  &  &  &  \\
0 & 0 & 0 & 0 & 0 & 0 & 0 &  &  &  \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &  &  \\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 &  \\
8 & 8 & 7 & 6 & 5 & 4 & 3 & 2 & 1 & 0 \\
\end{array}\right] + c_{8,8}\left[\begin{array}{cccccccccc}
0 &  &  &  &  &  &  &  &  &  \\
0 & 0 &  &  &  &  &  &  &  &  \\
0 & 0 & 0 &  &  &  &  &  &  &  \\
0 & 0 & 0 & 0 &  &  &  &  &  &  \\
0 & 0 & 0 & 0 & 0 &  &  &  &  &  \\
0 & 0 & 0 & 0 & 0 & 0 &  &  &  &  \\
0 & 0 & 0 & 0 & 0 & 0 & 0 &  &  &  \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &  &  \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 &  \\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 \\
\end{array}\right] + c_{9,9}\left[\begin{array}{cccccccccc}
0 &  &  &  &  &  &  &  &  &  \\
0 & 0 &  &  &  &  &  &  &  &  \\
0 & 0 & 0 &  &  &  &  &  &  &  \\
0 & 0 & 0 & 0 &  &  &  &  &  &  \\
0 & 0 & 0 & 0 & 0 &  &  &  &  &  \\
0 & 0 & 0 & 0 & 0 & 0 &  &  &  &  \\
0 & 0 & 0 & 0 & 0 & 0 & 0 &  &  &  \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &  &  \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &  \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
\end{array}\right]
\end{split}
\label{eq:complete:expansion:catalan:variables:on:main:diagonal}
\end{equation}
\begin{equation}
\text{Buondary conditions on free variables:}\quad
\left \{ c_{1,1} : c_{0,0}, \quad c_{2,2} : c_{0,0}, \quad c_{3,3} : c_{0,0}, \quad c_{4,4} : c_{0,0}, \quad c_{5,5} : c_{0,0}, \quad c_{6,6} : c_{0,0}, \quad c_{7,7} : c_{0,0}, \quad c_{8,8} : c_{0,0}, \quad c_{9,9} : c_{0,0}\right \}
\label{eq:complete:dependencies:catalan:variables:on:main:diagonal}
\end{equation}
\end{sidewaystable}

\subsubsection{Free variables lying on last but one row}

In \autoref{eq:complete:splitting:catalan:variables:on:last:but:one},
\autoref{eq:complete:expansion:catalan:variables:on:last:but:one} and
\autoref{eq:complete:dependencies:catalan:variables:on:last:but:one}
we report splitting, matrix expansion and dependencies of $\mathcal{C}$
using variables $c_{9,0},c_{9,1},\ldots, c_{9,9}$, respectively.

\begin{sidewaystable}
\scriptsize
\begin{equation}
\left[\begin{array}{cccccccccc}
c_{9,9} &  &  &  &  &  &  &  &  &  \\
c_{9,8} - 8 c_{9,9} & c_{9,9} &  &  &  &  &  &  &  &  \\
c_{9,7} - 7 c_{9,8} + 21 c_{9,9} & c_{9,8} - 7 c_{9,9} & c_{9,9} &  &  &  &  &  &  &  \\
c_{9,6} - 6 c_{9,7} + 15 c_{9,8} - 20 c_{9,9} & c_{9,7} - 6 c_{9,8} + 15 c_{9,9} & c_{9,8} - 6 c_{9,9} & c_{9,9} &  &  &  &  &  &  \\
c_{9,5} - 5 c_{9,6} + 10 c_{9,7} - 10 c_{9,8} + 5 c_{9,9} & c_{9,6} - 5 c_{9,7} + 10 c_{9,8} - 10 c_{9,9} & c_{9,7} - 5 c_{9,8} + 10 c_{9,9} & c_{9,8} - 5 c_{9,9} & c_{9,9} &  &  &  &  &  \\
c_{9,4} - 4 c_{9,5} + 6 c_{9,6} - 4 c_{9,7} + c_{9,8} & c_{9,5} - 4 c_{9,6} + 6 c_{9,7} - 4 c_{9,8} + c_{9,9} & c_{9,6} - 4 c_{9,7} + 6 c_{9,8} - 4 c_{9,9} & c_{9,7} - 4 c_{9,8} + 6 c_{9,9} & c_{9,8} - 4 c_{9,9} & c_{9,9} &  &  &  &  \\
c_{9,3} - 3 c_{9,4} + 3 c_{9,5} - c_{9,6} & c_{9,4} - 3 c_{9,5} + 3 c_{9,6} - c_{9,7} & c_{9,5} - 3 c_{9,6} + 3 c_{9,7} - c_{9,8} & c_{9,6} - 3 c_{9,7} + 3 c_{9,8} - c_{9,9} & c_{9,7} - 3 c_{9,8} + 3 c_{9,9} & c_{9,8} - 3 c_{9,9} & c_{9,9} &  &  &  \\
c_{9,2} - 2 c_{9,3} + c_{9,4} & c_{9,3} - 2 c_{9,4} + c_{9,5} & c_{9,4} - 2 c_{9,5} + c_{9,6} & c_{9,5} - 2 c_{9,6} + c_{9,7} & c_{9,6} - 2 c_{9,7} + c_{9,8} & c_{9,7} - 2 c_{9,8} + c_{9,9} & c_{9,8} - 2 c_{9,9} & c_{9,9} &  &  \\
c_{9,1} - c_{9,2} & c_{9,2} - c_{9,3} & c_{9,3} - c_{9,4} & c_{9,4} - c_{9,5} & c_{9,5} - c_{9,6} & c_{9,6} - c_{9,7} & c_{9,7} - c_{9,8} & c_{9,8} - c_{9,9} & c_{9,9} &  \\
\textcolor{blue}{c_{9,0}} & \textcolor{blue}{c_{9,1}} & \textcolor{blue}{c_{9,2}} & \textcolor{blue}{c_{9,3}} & \textcolor{blue}{c_{9,4}} & \textcolor{blue}{c_{9,5}} & \textcolor{blue}{c_{9,6}} & c_{9,7} & \textcolor{blue}{c_{9,8}} & \textcolor{blue}{c_{9,9}} \\
\end{array}\right]
\label{eq:complete:splitting:catalan:variables:on:last:but:one}
\end{equation}
\begin{equation}
    \begin{split}
    & c_{9,0}\left[\begin{array}{cccccccccc}
    0 &  &  &  &  &  &  &  &  &  \\
    0 & 0 &  &  &  &  &  &  &  &  \\
    0 & 0 & 0 &  &  &  &  &  &  &  \\
    0 & 0 & 0 & 0 &  &  &  &  &  &  \\
    0 & 0 & 0 & 0 & 0 &  &  &  &  &  \\
    0 & 0 & 0 & 0 & 0 & 0 &  &  &  &  \\
    0 & 0 & 0 & 0 & 0 & 0 & 0 &  &  &  \\
    0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &  &  \\
    0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &  \\
    1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
    \end{array}\right] + c_{9,1}\left[\begin{array}{cccccccccc}
    0 &  &  &  &  &  &  &  &  &  \\
    0 & 0 &  &  &  &  &  &  &  &  \\
    0 & 0 & 0 &  &  &  &  &  &  &  \\
    0 & 0 & 0 & 0 &  &  &  &  &  &  \\
    0 & 0 & 0 & 0 & 0 &  &  &  &  &  \\
    0 & 0 & 0 & 0 & 0 & 0 &  &  &  &  \\
    0 & 0 & 0 & 0 & 0 & 0 & 0 &  &  &  \\
    0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &  &  \\
    1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &  \\
    0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
    \end{array}\right] + c_{9,2}\left[\begin{array}{cccccccccc}
    0 &  &  &  &  &  &  &  &  &  \\
    0 & 0 &  &  &  &  &  &  &  &  \\
    0 & 0 & 0 &  &  &  &  &  &  &  \\
    0 & 0 & 0 & 0 &  &  &  &  &  &  \\
    0 & 0 & 0 & 0 & 0 &  &  &  &  &  \\
    0 & 0 & 0 & 0 & 0 & 0 &  &  &  &  \\
    0 & 0 & 0 & 0 & 0 & 0 & 0 &  &  &  \\
    1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &  &  \\
    -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &  \\
    0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
    \end{array}\right] + c_{9,3}\left[\begin{array}{cccccccccc}
    0 &  &  &  &  &  &  &  &  &  \\
    0 & 0 &  &  &  &  &  &  &  &  \\
    0 & 0 & 0 &  &  &  &  &  &  &  \\
    0 & 0 & 0 & 0 &  &  &  &  &  &  \\
    0 & 0 & 0 & 0 & 0 &  &  &  &  &  \\
    0 & 0 & 0 & 0 & 0 & 0 &  &  &  &  \\
    1 & 0 & 0 & 0 & 0 & 0 & 0 &  &  &  \\
    -2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 &  &  \\
    0 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 &  \\
    0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
    \end{array}\right] \\
    & c_{9,4}\left[\begin{array}{cccccccccc}
    0 &  &  &  &  &  &  &  &  &  \\
    0 & 0 &  &  &  &  &  &  &  &  \\
    0 & 0 & 0 &  &  &  &  &  &  &  \\
    0 & 0 & 0 & 0 &  &  &  &  &  &  \\
    0 & 0 & 0 & 0 & 0 &  &  &  &  &  \\
    1 & 0 & 0 & 0 & 0 & 0 &  &  &  &  \\
    -3 & 1 & 0 & 0 & 0 & 0 & 0 &  &  &  \\
    1 & -2 & 1 & 0 & 0 & 0 & 0 & 0 &  &  \\
    0 & 0 & -1 & 1 & 0 & 0 & 0 & 0 & 0 &  \\
    0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
    \end{array}\right] + c_{9,5}\left[\begin{array}{cccccccccc}
    0 &  &  &  &  &  &  &  &  &  \\
    0 & 0 &  &  &  &  &  &  &  &  \\
    0 & 0 & 0 &  &  &  &  &  &  &  \\
    0 & 0 & 0 & 0 &  &  &  &  &  &  \\
    1 & 0 & 0 & 0 & 0 &  &  &  &  &  \\
    -4 & 1 & 0 & 0 & 0 & 0 &  &  &  &  \\
    3 & -3 & 1 & 0 & 0 & 0 & 0 &  &  &  \\
    0 & 1 & -2 & 1 & 0 & 0 & 0 & 0 &  &  \\
    0 & 0 & 0 & -1 & 1 & 0 & 0 & 0 & 0 &  \\
    0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
    \end{array}\right] + c_{9,6}\left[\begin{array}{cccccccccc}
    0 &  &  &  &  &  &  &  &  &  \\
    0 & 0 &  &  &  &  &  &  &  &  \\
    0 & 0 & 0 &  &  &  &  &  &  &  \\
    1 & 0 & 0 & 0 &  &  &  &  &  &  \\
    -5 & 1 & 0 & 0 & 0 &  &  &  &  &  \\
    6 & -4 & 1 & 0 & 0 & 0 &  &  &  &  \\
    -1 & 3 & -3 & 1 & 0 & 0 & 0 &  &  &  \\
    0 & 0 & 1 & -2 & 1 & 0 & 0 & 0 &  &  \\
    0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 & 0 &  \\
    0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
    \end{array}\right]\\
    & + c_{9,7}\left[\begin{array}{cccccccccc}
    0 &  &  &  &  &  &  &  &  &  \\
    0 & 0 &  &  &  &  &  &  &  &  \\
    1 & 0 & 0 &  &  &  &  &  &  &  \\
    -6 & 1 & 0 & 0 &  &  &  &  &  &  \\
    10 & -5 & 1 & 0 & 0 &  &  &  &  &  \\
    -4 & 6 & -4 & 1 & 0 & 0 &  &  &  &  \\
    0 & -1 & 3 & -3 & 1 & 0 & 0 &  &  &  \\
    0 & 0 & 0 & 1 & -2 & 1 & 0 & 0 &  &  \\
    0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 &  \\
    0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
    \end{array}\right] + c_{9,8}\left[\begin{array}{cccccccccc}
    0 &  &  &  &  &  &  &  &  &  \\
    1 & 0 &  &  &  &  &  &  &  &  \\
    -7 & 1 & 0 &  &  &  &  &  &  &  \\
    15 & -6 & 1 & 0 &  &  &  &  &  &  \\
    -10 & 10 & -5 & 1 & 0 &  &  &  &  &  \\
    1 & -4 & 6 & -4 & 1 & 0 &  &  &  &  \\
    0 & 0 & -1 & 3 & -3 & 1 & 0 &  &  &  \\
    0 & 0 & 0 & 0 & 1 & -2 & 1 & 0 &  &  \\
    0 & 0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 &  \\
    0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
    \end{array}\right] + c_{9,9}\left[\begin{array}{cccccccccc}
    1 &  &  &  &  &  &  &  &  &  \\
    -8 & 1 &  &  &  &  &  &  &  &  \\
    21 & -7 & 1 &  &  &  &  &  &  &  \\
    -20 & 15 & -6 & 1 &  &  &  &  &  &  \\
    5 & -10 & 10 & -5 & 1 &  &  &  &  &  \\
    0 & 1 & -4 & 6 & -4 & 1 &  &  &  &  \\
    0 & 0 & 0 & -1 & 3 & -3 & 1 &  &  &  \\
    0 & 0 & 0 & 0 & 0 & 1 & -2 & 1 &  &  \\
    0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 1 &  \\
    0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
    \end{array}\right]
    \end{split}
\label{eq:complete:expansion:catalan:variables:on:last:but:one}
\end{equation}
\begin{equation}
\text{Buondary conditions on free variables:}\quad
\left \{ c_{9,0} : 4862 c_{0,0}, \quad c_{9,1} : 4862 c_{0,0}, \quad c_{9,2} : 3432 c_{0,0}, \quad c_{9,3} : 2002 c_{0,0}, \quad c_{9,4} : 1001 c_{0,0}, \quad c_{9,5} : 429 c_{0,0}, \quad c_{9,6} : 154 c_{0,0}, \quad c_{9,7} : 44 c_{0,0}, \quad c_{9,8} : 9 c_{0,0}, \quad c_{9,9} : c_{0,0}\right \}
\label{eq:complete:dependencies:catalan:variables:on:last:but:one}
\end{equation}
\end{sidewaystable}


%&LaTeX

\section{Feedforward Filters}

This lab covers the basic concepts of filtering and feedforward filters.
You may have also heard of feedforward filters referred to as
finite impulse response (FIR) filters. In this lab, we will cover the basic idea
of a filter, its mathematical representation (such as the defining equation,
frequency response, and transfer function), the relationship among
filter coefficients, zero placement, and filter type (low pass, high
pass, band reject), and some basic properties of filters.

\subsection{Overview of Filtering and J-DSP}

A \emph{digital filter} is a signal processing operation that can be
  described equivalently by its \emph{defining
  equation}, \emph{transfer function}, or \emph{frequency response}.
  Each representation completely defines the filter. It is advantageous
  to use each of the different representations depending on whether you are
  \emph{implementing}, \emph{analyzing}, or \emph{designing} an FIR filter:
\begin{eqnarray*}
  y[n] &=& \underbrace{\sum_{k=0}^M b_k x[n-k]}_{\text{defining equation}} \\
  \hline \\
  Y &=& H(z) X\\
  H(z) &=& \underbrace{\sum_{k=0}^M b_k z^{-k}}_{ \text{transfer function} } \\
  \hline \\
  Y &=&  \mathcal{H}({\hat{\omega}}) X \\
  H(e^{j\hat{\omega}})  = \mathcal{H}({\hat{\omega}})
  &=& \underbrace{\sum_{k=0}^M b_k e^{-j \hat{\omega}k}}_{\text{frequency response}}
\end{eqnarray*}

In these equations, $x[n]$ and $y[n]$ are the $n^\mathrm{th}$ samples
from the input and output, respectively, while $X$ and $Y$ represent
the entire input and output signal (all of the samples in the
signal). A $k$-sample time delay of a signal is produced by
multiplication by the delay operator, $z^{-1} = e^{-j
  \hat{\omega}k}$. In all three cases (but most simply for the
transfer function), we can obtain insight into the filter's operation
from the \emph{coefficients}, $b_k$. We can do this by factoring the
transfer function polynomial: its roots are the \emph{zeros} of the
filter and they can be real or complex.  The placement of the zeros in
the complex plane (most usefully expressed in polar coordinates) will
tell us which frequencies are suppressed and to what extent those
frequencies are suppressed (we can calculate each using the angle and
magnitude of the zero, respectively).

\begin{figure}
  \begin{center}
    \includegraphics[width=5in]{lab4/polezerodiagrams}
  \end{center}
  \caption{J-DSP setup for specifying filter coefficients and monitoring
  frequency response and zero location. (Left) Using the \block{coeff} block. (Right) Using the \block{PZ-Placement} block
  \label{fg:PZfigs}}
\end{figure}

J-DSP has a set of blocks that directly correspond to the concepts we
have learned. These include the \block{Filter} block, which takes an
input signal on the left and parameters (zero locations or
coefficients) at the bottom, and produces an output signal on the
right and can have its characteristics (frequency response or zero
locations) monitored via a connection from the top. We will use
\block{Filter} in two different ways:
\begin{enumerate}
\item We will specify the coefficients of the transfer function or
  generating equation using the \block{Coeff.} block and monitor the
  filter's output (using \block{Plot}), frequency response (using
  \block{Freq-Resp}), and zero placement (using \block{PZ-Plot}), as
  in Figure~\ref{fg:PZfigs} (left).
\item Instead of defining the filter using the \block{Coeff} block, we will
  place zeros directly in the complex plane using the \block{PZ
    Placement} block (which has an option to show the corresponding
  filter coefficients), as shown in Figure~\ref{fg:PZfigs} (right).
\end{enumerate}
Remember, we can define a specific filter using either of the methods
above. Sometimes it is easier to understand the filter using zeros and
sometimes it is easier to use the coefficients directly.  Either
method can be used to represent the same filter.

\subsubsection{From Filter Coefficients to Transfer Function and Frequency Response}

Given the coefficients of an FIR filter we can solve for the zero
locations and the frequency response.  For example the two-point
averaging system is given by:
\[
y[n] = \frac{1}{2}x[n] + \frac{1}{2}x[n-1]
\]
we can find the transfer function by rewriting the filter using the
delay operator, $z$:
\begin{eqnarray}
  Y    & = & \frac{1}{2} X + \frac{1}{2} z^{-1} X \\
  H(z) & = & \frac{Y}{X} = \frac{1}{2} (1 +  z^{-1})
\end{eqnarray}
If we're interested in the \emph{zero location} we can then multiply
$H(z)$ by $z/z$ to obtain:
\begin{equation}
  H(z) = \frac{\frac{1}{2} (z + 1)}{z}
\end{equation}
The root of the numerator, $z=-1$ is the location of the only zero
(the root(s) of the denominator for an FIR filter are always at $z=0$,
and do not affect the frequency response). We can also derive the frequency response from this by remembering that
$H(e^{j\hat{\omega}})  = \mathcal{H}({\hat{\omega}})$ or that $z=e^{j\hat{\omega}}$. From the zero
location, $z=-1$, we can immediately tell that the frequency response is zero at $e^{j\hat{\omega}}=-1$ or
$\hat{\omega}=\pi$. With a zero at an angle of
$\hat{\omega}=\pi$, this is a low-pass filter. As a warm-up, use the
J-DSP setup of Figure~\ref{fg:PZfigs} (left) to verify the expected zero
placement and frequency response.

\subsubsection{From Zero Placement to Filter Coefficients}

When we are given the zero placement, we can very easily determine the
filter coefficients because those zeros are the roots of a factored
polynomial. For example, given two complex conjugate zeros, $z_1$ and
$z_2$ (i.e., the real parts are equal, $\Real[z_{1}] = \Real[z_{2}] =
\Real[z_{1,2}] $, and the imaginary parts are negatives of one another
$ \Imag[z_{1}] = -\Imag[z_{2}] $ or, equivalently in polar
coordinates, $z_{1} = r e^{j\hat{\omega_0}}$ and $z_{2} = r
e^{-j\hat{\omega_0}}$), the transfer function is:
\begin{eqnarray}
  H(z) & = & (z - z_1)(z - z_2)/z^2 \\
  & = & (z^2 - (z_1 + z_2) z + z_1 z_2)/z^2 \\
  & = & 1 - 2 \Real[{z_{1,2}}] z^{-1} + r^2 z^{-2} \\
  & = & 1 - 2 \Real[r(\cos(\omega_0) \pm j\sin(\omega_0))] z^{-1} + r^2 z^{-2} \\
  & = & \underbrace{1}_{b_0} -\underbrace{2 r\cos(\omega_0)}_{b_1} z^{-1} + \underbrace{r^2}_{b_2} z^{-2}
\end{eqnarray}
At this point, we can rewrite the transfer function as the filter's
generating equation, using the delay operator $z^{-k}$, $y[n] = x[n] -
2 r\cos(\omega_0)x[n-1] + r^2 x[n-2]$. This allows us to read off the
filter coefficients: $b_0 = 1$, $b_1 = -2 r\cos(\omega_0)$, and $b_2 =
r^2$.

\subsection{Frequency Response and Pole-Zero Plots}

\paragraph{Step 1.1} Consider a filter that computes a running average
of three points of our input signal (a \emph{three-point averager}):
\begin{equation}
y[n] = \frac{1}{3} \sum_{k=0}^2 x[n-k]
     = \frac{1}{3}x[n] + \frac{1}{3}x[n-1] + \frac{1}{3}x[n-2]
\end{equation}

\begin{enumerate}\renewcommand{\theenumi}{\alph{enumi}}
\item Draw a block diagram for this filter.

\item How many zeros will this filter have?

\item Find and sketch the zero locations using pencil and paper, then
  use J-DSP to verify this.

\item Sketch the magnitude of the frequency response as a function of
	$\hat{\omega}$ from the plot of pole locations and verify this using
	J-DSP. How does the minimum of the magnitude of the frequency
	response relate to the polar representation of the zero locations?
	What kind of filter would you say this is?

\end{enumerate}

\begin{figure}
  \begin{center}
    \includegraphics[width=6in]{lab4/filteredsignal}
  \end{center}
\caption{J-DSP setup for Step 1.2 (e, f, and g). \label{fg:filtsig}}
\end{figure}

\paragraph{Step 1.2} A \emph{first-difference} filter is an
approximation to a discrete derivative operation. Its defining
equation is:
\begin{equation}
  y[n] = x[n] - x[n-1]
\end{equation}

\begin{enumerate}\renewcommand{\theenumi}{\alph{enumi}}

\item Draw a block diagram for this filter.

\item Derive the transfer function, $H(z)$, for this filter. From
	  this, determine the expression for the frequency response,
	  $\mathcal{H}(\hat{\omega})=H(e^{j\hat{\omega}})$.

\item From the transfer function, determine the filter's zero
	locations and sketch them. Check your results with J-DSP.

\item From the zero plot, sketch the magnitude of the filter's
	  frequency response as a function of $\hat{\omega}$. Use J-DSP to
	  check your results. What kind of filter would you say this is?

\item Use J-DSP to simulate this filter's response to the following
	  input. Set the \block{Sig Gen} to produce a sinusoid with
	  \option{frequency} $0.125\pi$, \option{gain} 1, \option{pulsewidth}
	  40. The diagram is shown in Figure \ref{fg:filtsig}.

\item Examine the plots of $X$ and $Y$ in J-DSP. Note that $Y$ appears
	  to be a scaled and shifted sinusoid of the same frequency as
	  $X$. The exception is the first point, $y[0]$. Explain why $y[0]$ is
	  different.

\item Estimate the frequency, amplitude, and phase of $Y$ directly
	  from its plot (ignoring $y[0]$).

\item To compare these measurements to theory, use your expression for
	  the filter's frequency response to calculate the amplitude and phase
	  at a frequency of $\hat{\omega} = \pi/8$. How do these compare to
	  what you determined from the J-DSP plots?

\end{enumerate}

\paragraph{Step 1.3} Just as we can compute a discrete first
derivative with a first-difference filter, we can compute a discrete
second derivative with a \emph{second difference filter}.

\begin{enumerate}\renewcommand{\theenumi}{\alph{enumi}}
\item Use your expression for the transfer function of the first
	  difference filter and your knowledge that the combined transfer
	  function of two filters cascaded, or connected in series, is the
	  product of their individual transfer functions to determine the
	  transfer function for a second-difference filter.

\item Draw a block diagram for this filter.

\item Determine the filter's zero locations and sketch them. Check
	  your results using J-DSP.

\item From the zero plot, sketch the magnitude of the filter's
	  frequency response as a function of $\hat{\omega}$. Use J-DSP to
	  check your results. What kind of filter would you say this is?

\end{enumerate}

\paragraph{Step 1.4} Consider a feedforward filter with complex
conjugate zeros at $z_{1,2} = -0.5 \pm j 0.5$.

\begin{enumerate}\renewcommand{\theenumi}{\alph{enumi}}
\item Determine the filter coefficients.

\item Use J-DSP to plot the frequency response of the filter.

\item What are the effects of the zeros on the frequency response?
	  What kind of filter would you call this?

\end{enumerate}

\subsection{Linearity and Cascading Filters}

\paragraph{Step 2.1} A system is called \emph{linear} if a sum of
different inputs produces an output that is the sum of the outputs for
the inputs taken individually.  Perform a simple test of the linearity
of this filter by doubling the input amplitude in J-DSP ($X' = 2X = X
+ X$). How does the new output amplitude compare to the old one?

\paragraph{Step 2.2} In one of the self-test exercises in the class notes,
	two filters with transfer functions $H_1(z) = b_0 + b_1z^{-1}$ and
	$H_2(z) = b'_0 + b'_1z^{-1}$ were connected in series, and it was
	shown that they could be connected in either order to produce the same
	composite effect (the same overall transfer function). Redo this
	exercise using the \emph{defining equations} for the two filters, i.e.,
	$y_1[n] = F_1(x[n])$ for the filter with transfer function $H_1(z)$
	and $y_2[n] = F_2(x[n])$ for the filter with transfer function
	$H_2(z)$. In other words, show that $F_2(F_1(x[n])) =
	F_1(F_2(x[n]))$.

\paragraph{Step 2.3} Use J-DSP to implement a system in which the output of
	the \block{Sig Gen} is connected to the three-point averager of step
	1.1, the output of which is in turn connected to the first-difference
	filter of step 1.2. Set the \block{Sig Gen} to output a periodic,
	rectangular waveform with amplitude of 1, pulse width of 10, and
	period of 20. This is a 50\% duty cycle square wave. Plot the final
	output waveform. What does it look like?

\paragraph{Step 2.4} Change your J-DSP simulation so that the first
	difference filter is first and the three-point averager is
	second. Plot the output. How does the output of this configuration
	compare to that of step 3.3?

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