Hook Lengths and 3-Cores

Recently, the first author generalized a formula of Nekrasov and Okounkov which gives a combinatorial formula, in terms of hook lengths of partitions, for the coefficients of certain power series. In the course of this investigation, he conjectured that a(n) = 0 if and only if b(n) = 0, where integers a(n) and b(n) are defined by Sigma(infinity)(n=0)a(n)x(n) : = Pi(infinity)(n=1)(1 - x(n))(8), Sigma(infinity)(n=0)b(n)x(n) : = Pi(infinity)(n=1)(1 - x(3n)3)/1 - x(n), The numbers a(n) are given in terms of hook lengths of partitions, while b(n) equals the number of 3-core partitions of n. Here we prove this conjecture.