Generating function for certain partition functions.

Question

what is the generating function of partitions of n into positive integers where the integers come in two kinds and the second kind has weight 23 times the weight of the first kind. Does the coefficients occurring here are well known or is it possible to write first (say) twenty coefficients easily?

Thanks for your time.

Have a great day.

Answer 1

I presume you want $P(q)P(q^{23})$ where $P(q)$ is the GF for conventional partitions, that is $$\prod_{n=1}^\infty\frac1{(1-q^n)(1-q^{23n})}.$$