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Search: id:A100818
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| A100818 |
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For a given unrestricted partition pi, let P(pi)=lambda(pi), if mu(pi)=0. If mu(pi)>0 then let P(pi)=nu(pi), where nu(pi) is the number of parts of pi greater than mu(pi), mu(pi) is the number of ones in pi, and lambda(pi) is the largest part of pi. Note that this is very similar to the "crank" of Andrews and Garvan. The number of partitions pi with P(pi) odd is the given sequence. |
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+0
4
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| 1, 2, 1, 4, 3, 8, 7, 15, 15, 27, 29, 48, 53, 82, 94, 137, 160, 225, 265, 362, 430, 572, 683, 892, 1066, 1370, 1640, 2078, 2487, 3117, 3725, 4624, 5519, 6791, 8092, 9885, 11752, 14263, 16922, 20416, 24167, 29007, 34254, 40921, 48213, 57345, 67409, 79864
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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The sequence is the same as A087787 except for the value of a(1) (this was established by George Andrews, Jan 18 2005). If "even" is replace by "odd" in the definition of the sequence, the new sequence is almost identical except for two values and a shift to the right.
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REFERENCES
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G. E. Andrews and F. G. Garvan, Dyson's crank of a partition, Bull. Amer. Math. Soc.18 (1988), 167-171
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FORMULA
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G.f.: x+(1/(1+x))* Product_{n>1}(1/(1-x^n))
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EXAMPLE
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a(3)=1 because P(3)=3, P(2 1)=1, and P(1 1 1)=0.
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MATHEMATICA
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Rest[ CoefficientList[ Series[x + 1/(1 + x) Product[1/(1 - x^n), {n, 50}], {x, 0, 50}], x]] (from Robert G. Wilson v Feb 11 2005)
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CROSSREFS
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Adjacent sequences: A100815 A100816 A100817 this_sequence A100819 A100820 A100821
Sequence in context: A112387 A076077 A087787 this_sequence A005291 A106624 A106622
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KEYWORD
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nonn
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AUTHOR
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David Newman (DavidSNewman(AT)hotmail.com), Jan 13 2005
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EXTENSIONS
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More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Feb 11 2005
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