Pentagonal Number Theorem

Euler's pentagonal number theorem gives a series expansion for the infinite product (1 minus x)(1 minus x squared)(1 minus x cubed) and so on, showing it equals the sum, over all integers k, of negative one to the k times x raised to the generalized pentagonal number k(3k minus 1)/2. Leonhard Euler discovered the identity, and F. Franklin later gave a bijective proof of it in 1881. The theorem is notable for the extensive cancellation it produces in the product's expansion, and it yields an efficient recurrence for computing the partition function p(n), since the apparently infinite series on its right side has only finitely many nonzero terms; it also arises as a special case of the Jacobi triple product and connects to modular forms and the Dedekind eta function. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/