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This function tends to exhibit deceptive behavior.
:p(1,4) = 5
:p(2,8) = 7
:p(3,12) = 9
:p(4,16) = 11
:p(5,20) = 13
:p(6,24) = '''16'''
Our original function p(<i>n</i>) is just p(1,<i>n</i>).
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The <i>x</i><sup>m</sup> term in this product counts the number of ways to write
<i>n</i> = <i>a</i><sub>1</sub> + 2<i>a</i><sub>2</sub> + 3<i>a</i><sub>3</sub> + ... = (1 + 1 + 1 + 1 + ...) + (2 + 2 + 2 + 2 + ...) + (3 + 3 + 3 ...) + ...,
where each number <i>i</i> appears <i>a</i><sub>i</sub> times. This is precisely the definition of a partition of <i>n</i>, so our product is the desired generating function. More generally, the generating function for the partitions of <i>n</i> into numbers from a set <i>A</i> can be found by taking only those terms in the product where <i>k</i> is an element of <i>A</i> This result is due to [[Leonhard Euler|Euler]]
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The formulation of the generating function is similar to the product formulation of many [[modular form|modular forms]], giving some idea of the connection between the two. It can also be used in conjunction with the [[pentagonal number theorem]] to derive a recurrence for the partition function stating that
p(<i>k</i>)
where the sum is taken over all [[polygonal number|pentagonal numbers]] of the form ½
Some values of the partition function are as follows:
*p(1) = 1
*p(2) = 2
*p(3) = 3
*p(4) = 5
*p(5) = 7
*p(6) = 11
*p(7) = 15
*p(8) = 22
*p(9) = 30
*p(10) = 42
*p(100) = 190569292
*p(1000) = 24061467864032622473692149727991
*p(10000) = 36167251325636293988820471890953695495016030339315650422081868605887952568754066420592310556052906916435144
==External Links==
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