Sum over k of n+k Choose m+2k by 2k Choose k by -1^k over k+1

Theorem
Let $m, n \in \Z_{\ge 0}$.

Then:
 * $\displaystyle \sum_k \binom {n + k} {m + 2 k} \binom {2 k} k \frac {\left({-1}\right)^k} {k + 1} = \binom {n - 1} {m - 1}$

where $\dbinom {n + k} {m + 2 k}$ etc. are binomial coefficients.