I am investigating the perturbation of the Jordan canonical form. In my work I must calculate the number of ways to factor $p^ {n-k} q^k$ where $p$ and $q$ are distinct primes (https://oeis.org/A054225). This sequence is generated by the function: $$ \prod_{i=1}^t \prod_{j=0}^i \frac{1}{1-x^iy^j}=1+x+xy+2x^2+2x^2y+2x^2y^2+3x^3+4x^3y+4x^3y^2+3x^3y^3+\ldots. $$ I've never solved a multivariable generating function before. Could you advise please, how to prove that the function is generated for the sequence, or advise the literature on this subject?

Any help would be greatly appreciated.