The following identity can be proved us generating functions and Newton's Binomial Theorem \begin{equation*} 2^{2n} = \sum_{k = 0}^{n} \binom{2k}{k} \binom{2n-2k}{n-k}, \qquad \text{for all $n \in \mathbb N$} \end{equation*} I have tried to get a combinatorial proof without success.
Does any one know such a proof?