New developments of an old identity

TL;DR

This paper provides a combinatorial proof of a well-known binomial identity, explores its generalizations, and derives the explicit generating function for a related binomial sequence.

Contribution

It offers a direct combinatorial proof of the identity, introduces two generalizations, and explicitly formulates the generating function for a binomial sequence.

Findings

Combinatorial proof of the identity using counting of subset pairs

Two new generalizations of the original identity

Explicit form of the generating function for the sequence inom{2n+k}{n}

Abstract

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 $k\in\mathds{R}$.