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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.

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You can find the answer in the paper Mark C. Wilson, Diagonal asymptotics for products of combinatorial classes: $$\sum_{k=0}^n\binom{n}{k}^d\sim\sqrt{\frac{2^{d-1}}{d}}\frac{2^{dn}}{(\pi n)^{\frac{d-1}{2} }}$$

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