I have to solve this discrete sum for a statistics question. This sum for
$$ \frac{1}{2^N} \sum_{d=-N,-(N-2),-(N-4)}^N \frac{\lvert d\rvert N!}{(\frac {N+d}{2})!(\frac {N-d}{2})!)}$$
To do this sum, I know that you have to split up the sum when N is odd or even but I don't know how to carry on. Do you need to understand the Gamma Function to do this sum?