Given $A,B \in \mathbb{Z}_+$ and $ 0 < t, q< 1$, I'd like to compute the coefficients $c_n(q,A,B)$ in the expansion of the product $$\prod_{i=0}^{A-1} \prod_{j=0}^{B-1} \frac{1}{1-t q^{i+j}} = \sum_{n=0}^{\infty} c_n t^n.$$ As $q \rightarrow 1$, this returns the well known formula $$\frac{1}{(1-t)^{AB}} = \sum_{n=0}^{\infty} \binom{AB+n-1}{n} t^n$$ which has a quick enumerative proof.

So far I've determined that the highest power of $q$ in $c_n(q,A,B)$ is $(A-1)(B-1)n$ less than the highest power of $q$ in the $q$-binomial coefficient $\binom{AB+n-1}{n}_q$. Can anyone see what these $c_n$ count in the expansion of the product? Any help would be greatly appreciated!