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

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

Then:
 * $\displaystyle \sum_k \binom {n + k} {2 k} \binom {2 k} k \frac {\paren {-1}^k} {k + 1} = \sqbrk {n = 0}$

where:
 * $\dbinom {n + k} {2 k}$ etc. are binomial coefficients
 * $\sqbrk {n = 0}$ is Iverson's convention.