of the identity (\[mi\]).

As a consequence of Theorem \[the.1\], we obtain the matrix identity (\[eq.5\]). Note that the matrix identity (\[eq2.2\]) is a special case of (\[eq.5\]) for $k=4$ and $t=1$, and the matrix identity (\[eq1.1\]) is also a special case of (\[eq.5\]) for $k=6$ and $t=2$.

An Identity of Cameron and Nkwanta
==================================

In their study of involutions in Riordan groups, Cameron and Nkwanta [@callan] obtained the following identity for $m\geq
0$, and asked for a purely combinatorial proof: $$\binom{n}{m}4^{n-m}=\sum_{k=0}^{n}\frac{k+1}{n+1}\binom{k+m+1}{k-m}\binom{2n+2}{n-k}.$$

It is clear that identity (\[eq.2\]) is a special case for $m=0$. To be consistent with our notation, we may rewrite the above identity in the following form: $$\label{cam}
\binom{i-1}{m}4^{i-1-m}=\sum_{j=1}^{i}\frac{j}{i}\binom{j+m}{2m+1}\binom{2i}{i-j}.$$

We now give a bijective proof (\[cam\]).

We recall that the number of partial $2$-Motzkin paths of length $i-1$ ending at level $j-1$ is given by $a_{i,j}=\frac{j}{i}\binom{2i}{i-j}$. We now consider the set of partial $2$-Motzkin paths with $m$ marked R-visible up steps and $m+1$ elevation lines such that there is exactly one marked R-visible up step between two adjacent elevation lines. We now have a combinatorial interpretation of the summand in (\[cam\]).

The summand in (\[cam\]) counts partial $2$-Motzkin paths of length $i-1$ ending at level $j-1$ with 