[Math] $P(x)=\sum_{n=0}^{\infty} p_nx^n=1+x+2x^2+3x^3+5x^4+7x^5+11x^6+\cdots$ and $Q(x)=\sum_{n=0}^{\infty} q_nx^n.$ What is $\dfrac{Q(x)}{P(x)}$

Question

Let
$$P(x)=\sum_{n=0}^{\infty} p_nx^n=1+x+2x^2+3x^3+5x^4+7x^5+11x^6+\cdots$$
be the partition generating function, and let $Q(x)=\sum_{n=0}^{\infty} q_nx^n$, where $q_n$ is the number of partitions of $n$ containing no $1$s.

Then $\dfrac{Q(x)}{P(x)}$ is a polynomial. What polynomial is it?

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What should I do? Should I plug the $P(x)$ into $Q(x)$? I'm a mess here, solutions are greatly appreciated!

Answer 1

Hint: Formally, we have $$P(x) = \prod_{k\ge1}\frac{1}{1-x^k}$$ (see e.g. Wikipedia). Can you do something similar for $Q(x)$?