What is the general formula for the product \begin{align} \prod_{m = 1}^{n - 1} \prod_{p = 1}^{n - 1} \left(1 - {z}^{m p}\right) = \sum_{n=0}^{N} a_n z^n\\ \end{align}
where $z \in \mathbb{C}$. (In particular what would be the $a_n$.) It bears a striking resemblance to a Fourier matrix. I first tried to develop a pattern via induction for the inner product with a given $m$, but I was unable to notice a simple relationship before the terms became too unruly.
Note: I already feel comfortable with proving convergence by taking logarithms and considering the double sum.
I believe that my goal is to evaluate the product in terms of q-Pochhamer symbols. I also looked at the 2015 A3 Putnam problem as it holds some similarities to the original product.