(85,4)(85.5,3)(86,4) (86,4)(86.5,5)(87,4) (87,4)(87.5,3)(88,4) (88,4)[(1,1)[4]{}]{} (92,8)[(1,0)[4]{}]{} (96,8)[(1,-1)[4]{}]{} (100,4)[(1,1)[8]{}]{} (108,12)[(1,-1)[8]{}]{} (116,4)[(1,1)[4]{}]{} (120,8)(1,0)[5]{}[(1,0)[0.4]{}]{} (123.85,8)[(1,-1)[4]{}]{} (124.15,8)[(1,-1)[4]{}]{}(128,4)[(1,1)[4]{}]{} (132,8)[(1,0)[4]{}]{} (80,8) (84,4) (88,4) (92,8) (96,8) (100,4) (104,8) (108,12) (112,8) (116,4) (120,8) (124,8) (128,4) (132,8) (136,8)

We conclude this section by giving a more general identity. Let $a_{i,j,k}$ be the number of partial $k$-Motzkin paths of length $i-1$ ending at level $j-1$. Then we have $$\label{cameron1}
\binom{i-1}{m}k^{i-1-m}=\sum_{j=1}^{i}a_{i,j,k-2}\binom{j+m}{2m+1}.$$

A Dyck path generalization of (\[eq1.1\])
=========================================

In this section, we give a Dyck path generalization of the matrix identity (\[eq1.1\]) on the little Schröder numbers. A $k$-Dyck path is a Dyck path in which an up step is colored by one of the $k$ colors $\{ 1, 2, \ldots, k\}$ if it not immediately followed by a down step . In this section, we aim to give the following generalization of (\[eq1.1\]).

Let $M=(m_{i,j})$ be a lower triangular matrix with the first column being the weight of $(t^2-t)$-Dyck paths of length $2i$. The other columns of $M$ are given by the following relation: $$\label{eq.6}
m_{i,j}=m_{i-1,j-1}+(t^2-t+1)m_{i-1,j}+(t^2-t)m_{i-1,j+1}.$$ Then we have the following matrix identity $$\label{eq.t}
(m_{i,j}) \times
\begin{bmatrix} 1 \\ t^2-(t-1)^2 \\     \ t^3-(t-1)^3 \\ \vdots
\end{bmatrix} = \begin{bmatrix} 1 \\ t^2+t \\   (t^2+t)^{2} \\
\vdots
\end{bmatrix}.$$

The matrix identity (\[eq1.1\]) is 