A combinatorial interpretation for the identity Sum_{k=0}^{n} binom{n}{k} Sum_{j=0}^{k} binom{k}{j}^{3}= Sum_{k=0}^{n} binom{n}{k}^{2}binom{2k}{k}

TL;DR

This paper provides a combinatorial interpretation of a binomial coefficient identity involving sums of binomial coefficients, connecting it to card deals, and offers a new perspective beyond algebraic proofs.

Contribution

It introduces a novel combinatorial interpretation for a known binomial identity, linking it to card deals, which was previously only proved algebraically.

Findings

Combinatorial interpretation in terms of card deals

Equivalence of two binomial sum expressions

Insight into binomial identities through combinatorics

Abstract

The title identity appeared as Problem 75-4, proposed by P. Barrucand, in Siam Review in 1975. The published solution equated constant terms in a suitable polynomial identity. Here we give a combinatorial interpretation in terms of card deals.