Number of Partitions as Coefficient of Power Series

Theorem
The number of partitions $\map p n$ of a (strictly) positive integer $n$ is equal to the coefficient of $x^n$ when the expression:
 * $\map f n = \dfrac 1 {\paren {1 - x} \paren {1 - x^2} \paren {1 - x^3} \cdots}$

is expanded into a power series.

That is:
 * $\map f n = 1 + \map p 1 x + \map p 2 x^2 + \map p 3 x^3 + \cdots$

or:
 * $\displaystyle \sum_{n \mathop \in \Z_{\ge 0} } \map p n q^n = \prod_{j \mathop \in \Z_{>0} } \dfrac 1 {1 - q^j}$

where $\map p 0 := 1$.