Question on generating function of integer partitions

Question

How can I show that $$\prod_{k \ge 1}(1+z^{2k}) = \prod_{k \ge 1}(1+z^k+z^{2k}+z^{3k}) \quad ?$$

I have worked on this for a while and I am even doubting that maybe both are not equal

Answer 1

The generating function for the partitions in which each part appears at most three times is $\prod_{k\geq 1}(1+x^k+x^{2k}+x^{3k})$, correct, but the generating function for the partitions in which every even part appears at most once is $$ \prod_{k\geq 1}(1+x^{2k})\prod_{k\geq 1}\frac{1}{1-x^{2k-1}} $$ since the odd parts may appear as many times as they want. Then we have to check that

$$\prod_{k\geq 1}\frac{1+x^{2k}}{1-x^{2k-1}}=\prod_{k\geq 1}\frac{1-x^{4k}}{1-x^k} $$ which is equivalent to $$ \prod_{k\geq 1}\frac{1-x^k}{1-x^{2k-1}}=\prod_{k\geq 1}\frac{1-x^{4k}}{1+x^{2k}} $$ or to $$ \prod_{k\geq 1}\frac{1-x^k}{1-x^{2k-1}}=\prod_{k\geq 1}(1-x^{2k}) $$ which is trivial since $$ \prod_{k\geq 1}(1-x^k) = \prod_{k\geq 1}(1-x^{2k})(1-x^{2k-1})$$ (the $k$ on the left is either odd or even).