New developments of an old identity

Abstract

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 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 $k\in\mathds{R}$.

8 pages

Cited by in corpus (2)

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