Higher Moments of Sums of Independent Random Variables Let $X_1 \dots X_n$ be independent random variables taking values $\{-1,1\}$ with equal probability 1/2. Let $S_n = \sum X_i$. Is there a closed form expression for $E[(S_n)^{2j}]$. If not a closed form expression then can we hope to get a nice tight upper bound. I am leaving tight unspecified here because I do not know myself how tight I want the bound to be so please tell me any non-trivial bounds. 

The random variable $S_n^{2j}$ takes the value $(2k-n)^{2j}$ with probability $\binom nk\frac 1{2^n}$, hence 
$$\mathbb E\left[S_n^{2j}\right]=\sum_{k=0}^n\binom nk(2k-n)^{2j}.$$
It involves computations of terms of the form $\sum_{k=0}^n\binom nk k^p$, $p\in\Bbb N$.
