Let $k$ be a fixed positive number and let $n$ go to infinity. Then by Polya (Problems and Theorems in Analysis) it holds that $$\sum_{k=0}^{n}\binom{n}{k}^t\simeq 2^{tn}\left(\frac{2}{\pi n}\right)^{\frac{t-1}{2}}\left(\frac{\pi}{2t}\right)^{\frac{1}{2}}.$$
I read the proof, but I did not understand it. Can anyone give a detailed proof?
Thank you
