# Identity for convolution of central binomial coefficients: \sum\limits_{k=0}^n \binom{2k}{k}\binom{2(n-k)}{n-k}=2^{2n}\sum\limits_{k=0}^n \binom{2k}{k}\binom{2(n-k)}{n-k}=2^{2n}

It’s not difficult to show that

On the other hand, we have $(1-z^2)^{-1}=\sum z^{2n}$. Squaring the first power series and comparing terms gives us

that is,

My question: is there a more direct, combinatorial proof of this identity? I’ve been racking my brains trying to come up with one but I’m not having much success.

It is possible to give a direct combinatorial proof, but it is quite difficult to find it.

One possibility is to use paths between points with integer coordinates and steps $(1,1)$ and $(1,-1)$.

1) $\binom{2n}{n}$ counts all paths from $(0,0)$ to $(2n,0)$.

2) $2^{2n}$ counts all paths starting from $(0,0)$ with $2n$ steps.

3) $\binom{2n}{n}$ counts all paths with $2n$ steps that never touch the $x$-axis again after the start. (This one is not obvious, but can be proved
with a bijection.)

Now you can conclude that all paths are a concatenation of a path that returns a certain number of times to the $x$-axis and a path that never does.

Note that the main difficulty here was that the two binomial coefficients are interpreted differently.