Let $S(n, t) = \sum\_{k = 0}^n {n \choose k} ^t$.


The task is to find asymptotic behavior of $S(n,5)$, $n \to \infty$.


Asymptotic for $S(n,0)$ and $S(n,1)$ is very simple.


For $S(n,2)$ we can use convolution for generating functions.


But, for case $n = 5$ I need help. 


Thank you for any help.



Asymptotic for binomial sums
You can find the answer in the paper [Mark C. Wilson, Diagonal asymptotics for products of combinatorial classes](http://www.cs.auckland.ac.nz/~mcw/Research/Outputs/Wils2013.pdf): $$\sum\_{k=0}^n\binom{n}{k}^d\sim\sqrt{\frac{2^{d-1}}{d}}\frac{2^{dn}}{(\pi
n)^{\frac{d-1}{2} }}$$


