Nonnegativity conditions for q-factorization exponents

Determine conditions on the coefficients r(n) of a power series with unit constant term that guarantee the q-factorization exponents a_n are nonnegative, thereby allowing the coefficients r(n) to admit an explicit counting interpretation.

Background

The paper factors a power series 1 + sum_{n>=1} r(n)q^n into a product of the form product_{n>=1}(1-q^n)^-a_n, and Theorem 2 gives an explicit formula for a_n in terms of the coefficients r(n). The exponents need not automatically be nonnegative. Establishing conditions on r(n) that ensure nonnegative exponents would make it possible to interpret the resulting product, and potentially the coefficients themselves, in terms of enumerative combinatorics.