Combinatorial proof of $\sum_{k} \binom{2r}{2k-1}\binom{k-1}{s-1}=2^{2r-2s+1}\binom{2r-s}{s-1}\ \ r,s\in \mathbb{N}_0$
Combinatorial proof of a binomial identity: $\sum_k \binom {2r} {2k-1}\binom{k-1}{s-1} = 2^{2r-2s+1}\binom{2r-s}{s-1}$