I've found this formula in a blog, it is in the answer to one question. But I don't know how to prove this:
Let $x_n$ be a sum of $n$ i.i.d. Bernoulli random variables with parameter $1/2$. Let $q\geq 2$. Show that $ E((2x_n-n)^q)=\sum_{k=0}^n{n \choose k}2^{-n}(2k-n)^q $
Thank you for your help.