  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
$k\in\mathds{R}$.
