Of course that is an infinite product that is hard to compute. It is hard to read off the coefficients especially for large n. Imagine asking specifically how many different partitions there are for 1000. It blows up pretty fast.
The above gives a pretty good approximation to its growth. For 1000 it is 1.4% off.
Still that's an estimate and exactly how good it is is questionable. There are also some infinite convergent sums, but that again has the same issues as infinite products.
In January, Ono said
"I can take any number, plug it into P, and instantly calculate the partitions of that number. P does not return gruesome numbers with infinitely many decimal places. It’s the finite, algebraic formula that we have all been looking for."
Apparently it is done with a "self similarity" of some specific subsequences. I'll just refer to the paper, but it is pretty clearly epic.
Edit: November 10, 2011
He gave a colloquium today. He explained the connection with Maass forms and mock theta functions.
http://en.wikipedia.org/wiki/Mock_modular_form
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