following matrix identity on Motzkin numbers:

$$\label{3p}
\begin{bmatrix} 1 \\1 & 1\\2 & 2 & 1\\4 & 5 & 3 & 1\\9 & 12 & 9 & 4 & 1\\& & \cdots &&&\ddots   \end{bmatrix}
\begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \\ 5 \\ \vdots
\end{bmatrix} = \begin{bmatrix} 1 \\ 3 \\ 3^2 \\ 3^3 \\ 3^4 \\ \vdots
\end{bmatrix},$$

where the first column is the sequence of Motzkin numbers, and matrix $A=(a_{ij})$ is generated by the following recurrence relation: $$a_{i,j}= a_{i-1, j-1} + a_{i-1, j} + a_{i-1, j+1}.$$ For $k=5$, we find the following matrix identity $$\label{5p}
\begin{bmatrix} 1 \\3 & 1\\10 & 6 & 1\\36 & 29 & 9 & 1\\137 & 132 & 57 & 12 & 1\\& & \cdots &&&\ddots  \end{bmatrix}
\begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \\ 5 \\ \vdots
\end{bmatrix} = \begin{bmatrix} 1 \\ 5 \\ 5^2 \\ 5^3 \\ 5^4 \\ \vdots
\end{bmatrix},$$ where the first column sequence A002212 in [@sloane], which has two interpretations, the number of $3$-Motzkin paths or the number of ways to assemble benzene rings into a tree [@hr]. Recall that a $3$-Motzkin path is a lattice path from $(0,0)$ to $(n-1,0)$ that does not go 