$m+1$ elevation lines such that there is exactly one marked up step between two adjacent elevation lines. Suppose that the $k$-th elevation line and the $k$-th marked up step of $P$ are at level $x_k$ and $y_k$, respectively. Such a configuration can be represented as follows: $$t_1\, | \, t_2 \, * \,  t_3\,  |\,  t_4\, * \,  t_5\, | \, \cdots \,  |
\,  t_{2m}  \,  * \, t_{2m+1} \,  |\,  t_{2m+2} ,$$ where $t_i$ denotes the numbers of unmarked R-visible up steps. It is clear that we have $t_1+t_2+\ldots+t_{2m+2}=j-1-m$, and the number of solutions of this equation equals the numbers of ways to distribute $j-1-m$ balls into $2m+2$ boxes when a box can have more than one ball. So this number equals the binomial coefficient ${ j+m \choose 2m+1}$.

We are now ready to give a combinatorial proof of the identity of Cameron and Nkwanta. We recall that a $2$-Motzkin path have two kind of horizontal steps, straight steps and wavy steps. We now need to introduce the third kind of horizontal steps – dotted steps. Therefore, the left hand side of (\[cam\]) is the number of free $3$-Motzkin paths with exactly $m$ dotted horizontal steps. We now 