interpretation of the entries in the matrix in (\[eq.5\]) is due to Cameron and Nkwanta [@cn]. A partial $(k,t)$-Motkzin path is defined as an initial segment of a $(k,t)$-Motkzin path. We say that a partial $(k, t)$-Motzkin path ends at level $j$ if its last step is at level $j$.

Let $m_{i,j}$ be the entries in the matrix in (\[eq.5\]). Then $m_{i,j}$ equals the the number of partial $(k-t-1, t)$-Motzkin paths of length $i-1$ that end at level $j-1$.

[*Proof.*]{} Regarding the first column of the matrix $M$, one sees that a partial $(k-t-1, t)$-Motzkin path that ends at level zero is just a $(k-t-1, t)$-Motzkin path. Let $a_{i,j}$ denote the number of partial $(k-t-1, t)$-Motzkin paths of length $i-1$ ending at level $j-1$. Let $P$ be a partial $(k-t-1,
t)$-Motzkin path of length $i-1$ that ends at level $j-1$ $(j>1)$. By considering the last step of $P$ and its weight, one sees that $a_{i,j}$ satisfies the recurrence relation (\[rc.5\]).

Let $P$ be a partial $(k-t-1, t)$-Motzkin path ending at level $j-1$. We need the notion of an [*elevated* ]{} partial Motzkin path, which has been introduced by Cameron and Nkwanta [@cn] in their combinatorial proof of the following identity 