below the $x$-axis and consists of up steps $U=(1,1)$, down steps $D=(1,-1)$, and three types of horizontal steps $H=(1,0)$. The above matrix $A=(a_{i,j})$ is generated by the first column and the following recurrence relation $$a_{i,j}= a_{i-1,j-1} + 3a_{i-1,j} +
a_{i-1,j+1}.$$ We may prove the above identities (\[3p\]) and (\[5p\]) by using method of Riordan arrays. So the natural question is to find a matrix identity for the sequence $(1, k, k^2, k^3, \ldots)$. We need the combinatorial interpretation of the entries in the matrix in terms of weighted partial Motzkin paths, as given by Cameron and Nkwanta [@cn]. To be precise, a partial Motzkin path, also called a Motzkin path from $(0,0)$ to $(n,k)$ in [@cn], is just a Motzkin path but without the requirement of ending on the x-axis. A weighted partial Motkzin a partial Motzkin path with the weight assignment that the horizontal steps are endowed with a weight $k$ and the down steps are endowed with a weight $t$, where $k$ and $t$ are regarded as positive integers. In this sense, our weighted Motzkin paths can be regarded as further generalization of $k$-Motzkin paths in the sense of $2$-Motzkin paths and $3$-Motkzin paths [@BdLPP; @deutschs; @sloane].

We 