Colored base-3 partitions, sequences of polynomials, and perfect numbers

Abstract: Motivated by the observation that the counting function of a certain base-3 colored partition contains the even perfect numbers as a subsequence, we begin by defining a sequence of polynomials in four variables and discuss their properties and combinatorial interpretations. We then concentrate on certain subsequences that are related to the Chebyshev polynomials of both kinds. Finally, we consider several sequences of single-variable polynomials that have meaningful combinatorial interpretations as well as interesting zero distributions.