the matrix identity (\[eq1.1\]). Let $g(x)$ be the generating function for the little Schröder numbers as given by $$\label{sg}
g(x)={1-3x-\sqrt{1-6x+x^2}\over 4x^2},$$ and let $f(x) =xg(x)$. Note that the generating function for the sequence $(1, 3, 7, 15, \ldots)$ equals $A(x)=\frac{1}{(1-x)(1-2x)}$. From the recurrence relation (\[r1.1\]) one may verify that the matrix in (\[eq1.1\]) is indeed the Riordan array $(g, xg)$. Therefore, the generating function for the right hand side of (\[eq1.1\]) equals $
g(x)A(xg(x))=\frac{1}{1-6x}$, which implies (\[eq1.1\]).

Using the same method, we can verify the matrix identity (\[3p\]) and (\[5p\]). Since we are going to establish a general bijection for weighted Motzkin numbers, here we omit the proofs.

Dyck path interpretation of (\[eq2.2\])
=======================================

In this section, we present a combinatorial interpretation of the matrix identity (\[eq2.2\]) by using Dyck paths. A [*Dyck path*]{} of length $2n$ is a path going from the origin $(0,0)$ to $(2n,0)$ using steps $U=(1,1)$ and down steps $D=(1,-1)$ such that no steps is below the $x$-axis [@De; @stanley]. The number of Dyck paths of length $2n$ equals the Catalan number $C_n$.

For a Dyck path $P$, the points on the $x$-axis except for the initial point are called return points. In this sense, the 