I am trying to prove the following equality (which has arised when I was trying to prove that $E|S_n| \rightarrow \sqrt{\frac{2n}{\pi}}$):
$ \sum_{x=1}^k \frac{x}{(k+x)!(k-x)! } = \frac{1}{2 \Gamma(k+1)\Gamma(k)}.$
I tried to expanding the left side different ways but the closest I got is this:
$\frac{1}{2 \Gamma(k+1)\Gamma(k)} = \frac{k}{2 (k!)^2}, \quad \sum_{x=1}^k \frac{x}{(k+x)!(k-x)! } = \frac{k}{(k!)^2}\left( 1\cdot\frac{1}{k+1} + 2\cdot\frac{(k-1)}{(k+1)(k+2)} +...+k \cdot \frac{(k-1)!}{(k+1)(k+2)...(2k)}\right).$
So, now I am stuck proving that latter sum is $\frac{1}{2}.$