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. 

For every fixed $n\geqslant1$,  $|S_n|\leqslant n$ almost surely, $P[|S_n|=n]=2^{1-n}$ and if $|S_n|\ne n$ then $|S_n|\leqslant n-1$, hence 
$$
\lim\limits_{j\to\infty}n^{-2j}\cdot E[(S_n)^{2j}]=P[|S_n|=n]=2^{1-n}.
$$
To prove this (or if nonasymptotic bounds are needed), use the inequalities
$$
P[|S_n|=n]\cdot n^{2j}\leqslant E[(S_n)^{2j}]\leqslant P[|S_n|=n]\cdot n^{2j}+(n-1)^{2j}.
$$
