Sum over k of r Choose k by s+k Choose n by -1^r-k/Corollary

Corollary to Sum over $k$ of $\dbinom r k \dbinom {s + k} n \left({-1}\right)^{r - k}$
Let $r \in \Z_{\ge 0}, n \in \Z$.

Then:
 * $\displaystyle \sum_k \binom r k \binom k n \left({-1}\right)^{r - k} = \delta_{n r}$

where $\delta_{n r}$ is the Kronecker delta.

Proof
From Sum over $k$ of $\dbinom r k \dbinom {s + k} n \left({-1}\right)^{r - k}$:
 * $\displaystyle \sum_k \binom r k \binom {s + k} n \left({-1}\right)^{r - k} = \binom s {n - r}$

which holds for $s \in \R, r \in \Z_{\ge 0}, n \in \Z$.

Setting $s = 0$:
 * $\displaystyle \sum_k \binom r k \binom k n \left({-1}\right)^{r - k} = \binom 0 {n - r}$

We have by definition of binomial coefficient that:
 * $\dbinom 0 0 = 1$

and:
 * $\forall n \in \Z_{\ne 0}: \dbinom 0 n = 0$

So, using Iverson's convention:
 * $\dbinom 0 {n - r} = \left[{n = r}\right]$

The result follows by definition of the Kronecker delta.