%0 Journal Article
%T New developments of an old identity
%A Rui Duarte
%A Ant¨Žnio Guedes de Oliveira
%J Mathematics
%D 2012
%I arXiv
%X We give a direct combinatorial proof of a famous identity, $$ \sum_{i+j=n} m{2i}{i} \binom{2j}{j} = 4^n $$ by actually counting pairs of $k$-subsets of $2k$-sets. Then we discuss two different generalizations of the identity, and end the paper by presenting in explicit form the ordinary generating function of the sequence $(\strut\binom{2n+k}{n})_{n\in\mathds{N}_0}$, where $k\in\mathds{R}$.
%U http://arxiv.org/abs/1203.5424v4