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

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!

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