New developments of an old identity

Details New developments of an old identity New developments of an old identity

2012-03-24

arxiv.org/abs/1203.5424v1

Mathematics

Combinatorics

7 pages

Scientific paper

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 $k\in\mathds{R}$.

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