Is there any formula for calculating $\sum_{k=0}^n {n\choose k} {2n\choose 2k}$ ?
One possible way is to use Stirling's approximation, but couldn't reach a reasonable answer.
$\endgroup$ 21 Answer
$\begingroup$This is probably not the most pleasant result. $$S_n=\sum_{k=0}^n {n\choose k} {2n\choose 2k}=\, _3F_2\left(\frac{1}{2}-n,-n,-n;\frac{1}{2},1;-1\right)$$ where appears the generalized hypergeometric function.
However, plotting $\log(S_n)$ as a function of $n$ reveals almost a straight line and a simple linear regression gives $$\log(S_n)\approx 2.06597 n-2.27715$$ ($R^2=0.999998$) which gives at least the order of magnitude.
$\endgroup$
