Author:]{} William Y. C. Chen, chen@nankai.edu.cn

Introduction
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This paper is motivated the following matrix identity obtained by Shapiro, Woan and Getu [@shapirotc] in their study of the moments of a Catalan triangle [@chapman; @shapiro; @sulanke]: $$\label{eq2.2}
\begin{bmatrix} 1 \\2 & 1\\5 & 4 & 1\\14 & 14 & 6 & 1\\42 & 48 & 27 & 8 & 1\\& & \cdots &&&\ddots  \end{bmatrix}
 \begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \\ 5 \\ \vdots
\end{bmatrix} = \begin{bmatrix} 1 \\ 4 \\ 4^2 \\ 4^3 \\ 4^4 \\ \vdots
\end{bmatrix},$$ where the first column of the first matrix is the Catalan number $C_n={1\over n+1} {2n \choose n}$ and $a_{i,j}$ (the entry in the $i$-th row and $j$-th column) is determined by the following recurrence relation for $j\geq 2 $: $$\label{r2.2}
a_{i,j}=a_{i-1, j-1}+2a_{i-1,j}+a_{i-1,j+1}   .$$ Another proof of the above identity is given by Woan, Shapiro and Rogers [@woan] while computing the areas of parallelo-polyominos via generating functions.

The first result of this paper is a combinatorial interpretation of the identity (\[eq2.2\]) in terms of Dyck paths.

One main objective of this paper is to give a matrix identity that extends the sequence $(1, 4, 4^2, 4^3, \ldots)$ to $(1, k, 