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The function that appears in the denominator in the third and fourth lines of the formula is the [[Euler function]]. The equality between the product on the first line and the formulas in the third and fourth lines is Euler's [[pentagonal number theorem]].
The exponents of <math>x</math> in these lines are the [[pentagonal number]]s <math>P_k = k(3k-1)/2</math> for <math>k \in \{ 0, 1,-1,2,-2,\dots\}</math> (generalized somewhat from the usual pentagonal numbers, which come from the same formula for the positive values {{nowrap|of <math>k</math>).}} The pattern of positive and negative signs in the third line comes from the term <math>(-1)^k</math> in the fourth line: even choices of <math>k</math> produce positive terms, and odd choices produce negative terms.
More generally, the generating function for the partitions of <math>n</math> into numbers selected from a set <math>A</math> of positive integers can be found by taking only those terms in the first product for which <math>k\in A</math>. This result is due to [[Leonhard Euler]].{{r|e}} The formulation of Euler's generating function is a special case of a [[q-Pochhammer symbol|<math>q</math>-Pochhammer symbol]] and is similar to the product formulation of many [[modular form]]s, and specifically the [[Dedekind eta function]].
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