Pentagonal number theorem

In mathematics, Euler's pentagonal number theorem relates the product and series representations of the Euler function. It states that

${\displaystyle \prod _{n=1}^{\infty }\left(1-x^{n}\right)=\sum _{k=-\infty }^{\infty }\left(-1\right)^{k}x^{k\left(3k-1\right)/2}=1+\sum _{k=1}^{\infty }(-1)^{k}\left(x^{k(3k+1)/2}+x^{k(3k-1)/2}\right).}$

In other words,

${\displaystyle (1-x)(1-x^{2})(1-x^{3})\cdots =1-x-x^{2}+x^{5}+x^{7}-x^{12}-x^{15}+x^{22}+x^{26}-\cdots .}$

The exponents 1, 2, 5, 7, 12, ... on the right hand side are given by the formula g_k = k(3k − 1)/2 for k = 1, −1, 2, −2, 3, ... and are called (generalized) pentagonal numbers (sequence A001318 in the OEIS). (The constant term 1 corresponds to ${\displaystyle k=0}$.) This holds as an identity of convergent power series for ${\displaystyle |x|<1}$, and also as an identity of formal power series.

A striking feature of this formula is the amount of cancellation in the expansion of the product.