Let $P(x)=\sum_{n=0}^{\infty} p_nx^n$ be the partition generating function, and let $P^*(x)=\sum_{n=0}^{\infty} p^*_nx^n$, where $$p^*_n = \binom{\text{number of partitions of }n}{\text{into an even number of parts}} - \binom{\text{number of partitions of }n}{\text{into an odd number of parts}}.$$

Compute the truncation of $P(x)P^*(x)$ to degree $10$; that is, determine the polynomial consisting of all terms in the power series expansion of $P(x)P^*(x)$ with degree less than or equal to $10$.

How should I approach this problem?

  • $\begingroup$ Can you write $P(x)$ as a product? And $P^\ast(x)$? $\endgroup$ – Daniel Fischer Jun 13 '15 at 20:34
  • $\begingroup$ I’ve not really thought about it, but it may be helpful to know that $(-1)^np_n^*$ is the number of partitions of $n$ into distinct odd parts; there are two proofs of this here. $\endgroup$ – Brian M. Scott Jun 14 '15 at 3:26
  • $\begingroup$ @BrianM.Scott's answer over there almost contains what I was aiming at. So I recommend looking at that. $\endgroup$ – Daniel Fischer Jun 14 '15 at 19:29

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