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where each number ''i'' appears ''a''<sub>''i''</sub> times. This is precisely the definition of a partition of ''n'', so our product is the desired generating function. More generally, the generating function for the partitions of ''n'' into numbers from a set ''A'' can be found by taking only those terms in the product where ''k'' is an element of ''A''. This result is due to [[Leonhard Euler|Euler]].
The formulation of the generating function is similar to the product formulation of many [[modular form
p(<i>k</i>) − p(''k'' − 1) − p(''k'' − 2) + p(''k'' − 5) + p(''k'' − 7) − p(''k'' − 12) − ... = 0,
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