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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.

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What does "solve" mean for you? You can't expect a simple formula. –  Brendan McKay Dec 1 '13 at 18:07
I'm not sure if this is what you want, but the oeis page also gives some programs (maple, haskell, pari, mathematica) to compute the elements of the sequence. –  guest Dec 1 '13 at 20:20
I would like to know the technique of proving that the sequence is generated by the function. –  Alexander Dec 2 '13 at 5:20

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