New developments of an old identity

Rui Duarte, António Guedes de Oliveira

We give a direct combinatorial proof of a famous identity, $$\sum _{i+j=n}m2ii\left(\genfrac{}{}{0ex}{}{2j}{j}\right)=4^n$$ by actually counting pairs of $k$-subsets of $2k$-sets. Then we discuss t…