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
$$\sum^{\infty}_{n=0}\, a(n)x^{n} := \prod^{\infty}_{n=1} \, (1-x^{n})^8,$$
$$\sum^{\infty}_{n=0} \, b(n)x^{n} := \prod^{\infty}_{n=1} \, \frac{(1-x^{3n})^{3}}{1-x^n} .$$
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.