Euler's Pentagonal Number Theorem

$$\prod_{n=1}^\infty (1-x^n) =\sum_{n=-\infty}^\infty (-1)^nx^{n(3n+1)/2},$$ (1)

where $n(3n+1)/2$ are generalized Pentagonal Numbers. Related equalities are

$$\prod_{k=1}^\infty (1-x^kt)=\sum_{n=0}^\infty {(-1)^nx^{n(n+1)/2}t^n\over \prod_{k=1}^n (1-x^k)}$$ (2)

$$\prod_{k=1}^\infty (1-x^kt)^{-1}=\sum_{n=0}^\infty {t^n\over \prod_{k=1}^n (1-x^k)}.$$ (3)

See also Partition Function P, Pentagonal Number