Euler's Pentagonal Numbers Theorem

Theorem
Consider the infinite product:
 * $\displaystyle P = \prod_{n \mathop \in \Z_{>0} } \left({1 - x^n}\right)$

Then $P$ can be expressed as:
 * $\displaystyle P = \sum_{n \mathop \in \Z_{>0} } \left({-1}\right)^{\left \lceil{n / 2}\right\rceil} x^{GP_n}$

where:
 * $\left \lceil{n / 2}\right\rceil$ denotes the ceiling of $n / 2$
 * $GP_n$ denotes the $n$th generalized pentagonal number.

That is:
 * $P = 1 - x - x^2 + x^5 + x^7 - x^{12} - x^{15} + \cdots$