High Quality Content by WIKIPEDIA articles! In mathematics, the pentagonal number theorem, originally due to Euler, relates the product and series representations of the Euler function. It states that prod_{n=1}^infty (1-x^n)=sum_{k=-infty}^infty(-1)^kx^{k(3k-1)/2}. In other words, (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. A striking feature of this expansion is the amount of cancellation in the product. The indices 1, 2, 5, 7, 12, ... appearing on the right hand side are called pentagonal numbers (or more accurately, generalized pentagonal numbers).

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