Combinatorics Catastrophe

Question

How will you solve

$$\sum_{i=1}^{n}{2i \choose i}\;?$$

I tried to use Coefficient Method but couldn't get it! Also I searched for Christmas Stocking Theorem but to no use ...

Answer 1

This is OEIS A006134; the generating function is

$$g(x)=\frac1{(1-x)\sqrt{1-4x}}\;,$$

but no closed form is given. There is an approximation

$$a(n)\sim\frac{2^{2n+2}}{3\sqrt{\pi n}}\;.$$

Correction: It’s one less than the sequence from OEIS, whose terms include $\binom00=1$.

Answer 2

Maple gives a "closed form" involving a hypergeometric function: $$ -1-{2\,n+2\choose n+1}\; {\mbox{$_2$F$_1$}(1,n+\frac32;\,n+2;\,4)}-\frac{i \sqrt {3}}{3} $$