Is there a combinatorial proof of the Taylor series $\frac1{\sqrt{1-4x}}=\sum_{n=0}^\infty{2n\choose n}x^n$?
Identity for convolution of central binomial coefficients: $\sum\limits_{k=0}^n \binom{2k}{k}\binom{2(n-k)}{n-k}=2^{2n}$