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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum^{\infty}_{n=0}\, a(n)x^{n} := \prod^{\infty}_{n=1} \, (1-x^{n})^8,$$\end{document}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum^{\infty}_{n=0} \, b(n)x^{n} := \prod^{\infty}_{n=1} \, \frac{(1-x^{3n})^{3}}{1-x^n} .$$\end{document}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.