generating function for the number of partitions of n in which each odd part can occur any number of times but each even part can occur at most twice

Question

Write down the generating function for the number of partitions of $n$ in which each odd part can occur any number of times but each even part can occur at most twice. Apply algebra to this generating function to complete the following theorem:

The number of partitions of n in which each odd part can occur any number of times but each even part can occur at most twice is the number of partitions of n in which...

The generating function I've found for the first part is $$\prod_{k\geq1}\frac{(1+x^{2k})^2}{1-x^{2k-1}}$$ but I'm unsure how to start the second part, any help would be appreciated.

Answer 1

The generating function for the first part is \begin{eqnarray*} \prod_{n=1}^{\infty} \frac{ (1+x^{2n}+x^{4n})}{1-x^{2n-1}}. \end{eqnarray*} The terms in the numerator deal with the even numbers occurring at most twice & the terms in the denominator deal with the odd numbers.

Now observe that \begin{eqnarray*} 1+x^{2n}+x^{4n} = \frac{ 1-x^{6n} }{ 1-x^{2n} }. \end{eqnarray*} The generating function can be rewritten as \begin{eqnarray*} \prod_{n \geq 1 \text{ and } 6 \nmid n }^{\infty} \frac{ 1}{1-x^{n}}. \end{eqnarray*} So it is the same as partitioning numbers whose parts are not multiples of $6$.