Combinatorial proof of $\sum\limits_{k=0}^n \binom{2k}{k} \binom{2n-2k}{n-k} =4^n$
Identity for convolution of central binomial coefficients: $\sum\limits_{k=0}^n \binom{2k}{k}\binom{2(n-k)}{n-k}=2^{2n}$