Number of partitions with distinct even parts/parts with multiplicity $\leq 3$

Question

I am supposed to solve a problem regarding partitions of $n \in \mathbb{N}$ into:

1. distinct even parts
2. parts with multiplicity $\leq 3$

I am supposed to prove that 1. and 2. are equal.

So I tried using the generating functions, and if I'm not mistaken they should look like this:

1. $\Pi_{k \geq 0} (1+x^{2k})$
2. $\Pi_{k \geq 0} (1+x^k+x^{2k}+x^{3k})$ or maybe $\Pi_{k \geq 0} (1-x^k+x^{2k}-x^{3k})$, if I want to distinguish the parts with even from the parts with odd multiplicity

But I can't see how the generating funtions have the equal series expansion. Any hints?

Answer 1

The sequence counting these partitions is OEIS A001935. "Distinct even parts" does not mean that all parts are even, it means that even parts are distinct and odd parts have no restriction. The corresponding generating function is $$\begin{align} \prod_{k=1}^\infty\frac{1+x^{2k}}{1-x^{2k-1}}&=\prod_{k=1}^\infty\frac{1-x^{4k}}{1-x^{2k-1}}\frac1{1-x^{2k}}\\ &=\prod_{k=1}^\infty\frac{1-x^{4k}}{1-x^k}\\ &=\prod_{k=1}^\infty(1+x^k+x^{2k}+x^{3k}) \end{align}$$ The last line is the generating function for partitions with multiplicity at most $3$.