Combinatorial analysis of integer power product expansions

Let \( f(x) = 1 + \sum_{n=1}^\infty a_n x^n \) be a formal power series with complex coefficients. Let \( \{r_n\}_{n=1}^\infty \) be a sequence of nonzero integers. The Integer Power Product Expansion of \( f(x) \), denoted ZPPE, is \( \prod_{k=1}^\infty (1 + w_k x^k)^{r_k} \). Integer Power Product Expansions enumerate partitions of multi-sets. The coefficients \( \{w_k\}_{k=1}^\infty \) themselves possess interesting algebraic structure. This algebraic structure provides a lower bound for the radius of convergence of the ZPPE and provides an asymptotic bound for the weights associated with the multi-sets.