k^2,
k^3, \ldots)$ in (\[eq2.2\]). The following matrix identity was proved by Cameron and Nkwanta [@cn] that arose in a study of elements of order $2$ in [*Riordan groups*]{} [@aigner; @shapiroba; @shapirotr; @sprugnoli]: $$\label{eq1.1}
\begin{bmatrix} 1 \\3 & 1\\11 & 6 & 1\\45 & 31 & 9 & 1\\197 & 156 & 60 & 12 & 1 & &\\& & \cdots &&&\ddots  \end{bmatrix}
\begin{bmatrix} 1 \\ 3 \\ 7 \\ 15 \\ 31 \\ \vdots
\end{bmatrix} = \begin{bmatrix} 1 \\ 6 \\ 6^2 \\ 6^3 \\ 6^4 \\
\vdots
\end{bmatrix},$$ where the entry $a_{i,j}$ ($i$th row and $j$th column) in the above matrix satisfies the recurrence relation $$\label{r1.1}
a_{i,j}=a_{i-1,j-1}+3a_{i-1,j}+2a_{i-1,j+1}$$ for $j\geq 2$ and the $a_{i,1}$ is the $i$-th [*little Schröder number*]{} $s_i$ (sequence A001003 in [@sloane]), which counts Schröder paths of length $2(i+1)$. A [*Schröder path*]{} is a lattice path starting at (0, 0) and ending at $(2n, 0)$ and using steps $H=(2, 0)$, $U=(1, 1)$ and $D=(1,
-1)$ such that no steps are below the $x$-axis and there are no peaks at level one. Imposing this last peak condition gives us little Schröder numbers while without it we would have the [*large Schröder numbers*]{}.

For $k=3$, we obtain the 