How does $P(n)$ (Partition Function P) work?

Question

I am trying to do Project Euler #78, which is about the different ways of splitting up coins into piles. I realized that this is just the number of integer partitions of the number of coins, specifically the Partition Function P. I am having trouble figuring out what the MathWorld page on the subject means for a generating function. I know it has to do with a polynomial and pentagonal number powers. Here is the generating function it gives:

$(q)_{\infty}\equiv \prod_{m=1}^{\infty}(1-q^{m})$

$=\sum_{-\infty}^{\infty}(-1)^{n}q^{n(3n+1)/2}$

$=1-q-q^{2}+q^{5}+q^{7}-q^{12}-q^{15}+q^{22}+q^{26}+...$

I would greatly appreciate if someone who understands this could help explain it to me and maybe give a worked out example for some small numbers. Thanks!

Answer 1

That's not a generating function for the partition function; the reciprocal of what you have written is a generating function for the partition function. But the people at Project Euler have told us that they don't want people asking for help at places like this, so I'm afraid you're on your own.

EDIT: Since the software has bumped this question up to the front page anyway, perhaps I could point out that OP's request for "a worked example for some small numbers" doesn't make much sense in the context of the question asked, since there is no sensible interpretation of the word "example" and no sensible place to insert any small numbers. There may be a reasonable question lurking here somewhere, maybe even one that wouldn't run afoul of Project Euler rules, but OP hasn't asked it.