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

Theorem

$\ds \sum_k \binom {n + k} {2 k} \binom {2 k} k \frac {\paren {-1}^k} {k + 1} = \sqbrk {n = 0}$

$\ds $

$\ds \sum_k \binom {n + k} {2 k} \binom {2 k} k \frac {\left({-1}\right)^k} {k + 1}$

$\ds $

$=$

$\ds \sum_k \binom {n + k} k \binom n k \frac {\left({-1}\right)} {k + 1}$

$\ds $

$=$

$\ds \sum_k \binom {n + k} k \binom {n + 1} {k + 1} \frac {\left({-1}\right)^k} {n + 1}$

$\ds $

$=$

$\ds \frac 1 {n + 1} \sum_k \binom {- \left({n + 1}\right)} k \binom {n + 1} {k + 1}$

$\ds $

$=$

$\ds \frac 1 {n + 1} \binom {n + 1 - \left({n + 1}\right)} {n + 1 - 1 + 0}$

$\ds $

$=$

$\ds \frac 1 {n + 1} \binom 0 n$

$\ds $

$=$

$\ds \left[{n = 0}\right]$

$\blacksquare$