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.


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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.