On periodic sign changes for weighted representations of integers as colored sums of triangular and generalized pentagonal numbers

In this paper, we study sign changes and vanishing for the number of representations of an integer as a sum of triangular numbers plus three-colored sums of generalized pentagonal numbers with an even number of parts minus those with an odd number of parts. We study this by investigating coefficients appearing in the $q$-series expansion of $F(z)=\frac{\eta (z)}{{\eta (2z)}^2{\eta (3z)}^3}$, where $\eta$ is the Dedekind-eta function. We use the Hardy-Ramanujan-Rademacher circle method to give an asymptotic formula for the coefficients.