Given that $$\prod_{j=1}^{\infty} (1-z^{j^2})$$ is of the form $1+\sum_{d=1}^{\infty} a_{d}z^{c_{d}}$
where $c_{1}<c_{2}<... \in \mathbb{N}, a_{d} \in \mathbb{Z}$. Is there a constant $B>0$ so that $|a_{d}| \leq B$.
Background:
$$\prod_{j=1}^{\infty}(1-z^{j}) = 1 + \sum_{k=1}^{\infty}(-1)^{k}(z^{\frac{k(3k-1)}{2}}+z^{\frac{k(3k+1)}{2}})$$
by the Pentagonal number theorem (https://en.wikipedia.org/wiki/Pentagonal_number_theorem).
This question has been asked to understand distinct square partitions with odd and even containers.