I am trying to work through exercise 14.5 from Tom M. Apostol's Introduction to Analytic Number Theory. There does exist a post already on stack exchange for the first part of this exercise. I am a bit confused about the second part of the exercise. Namely, it required proving this result
$$ \sum_{m=1}^{\infty} x^{m(m-1)/2} = \prod_{n=1}^\infty \frac{1-x^{2n}}{1-x^{2n-1}} $$
which I was succesfully able to do.
However, I have no idea how this proves Gauss' triangular number theorem. I am aware that the cube of the LHS provides a generating function for the number of ways to partition a number into three triangular numbers. However, I do not know how to show that all the coefficients of the cube of the power series in the RHS have only positive contributions.
I have noticed a few things. I can see that the RHS is the same as $$ \frac{\sum p_o(n)x^n}{\sum p_e(n)x^n} $$ where $p_o(n)$ is the number of ways to partition into odd numbers, and $p_e(n)$ is the number of ways to partition into even numbers. Aside from this, I have made virtually no progress. I would love to have some opinions as to how to tackle the problem.