Prove an equation about binomial coefficients Could we prove:
$ \sum_{k} \binom{2k}{k}\binom{n+k}{m+2k} \frac{(-1)^k}{k+1} = \binom{n-1}{m-1}$ when $m,n \in N$

When $n=1$ it's easy to evaluate for all $m\ge1$. Then it's easy to prove the $n+1$ by induction, given the $n$ case (for all $m$): just write
$$
\binom{n+k}{m+2k} = \binom{n-1+k}{m+2k} + \binom{n-1+k}{m-1+2k}.
$$
