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Search: id:A118083
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| A118083 |
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Number of partitions of n such that largest part k occurs at least floor(k/2) times. |
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3
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| 1, 1, 2, 3, 4, 5, 7, 8, 11, 13, 17, 20, 26, 30, 38, 45, 55, 64, 79, 91, 110, 128, 152, 176, 209, 240, 282, 325, 379, 434, 505, 576, 666, 760, 873, 993, 1139, 1290, 1473, 1668, 1897, 2141, 2430, 2736, 3095, 3481, 3925, 4404, 4958, 5550, 6232, 6968, 7805, 8710
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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Also number of partitions of n such that if the number of parts is k, then the smallest part is at least floor(k/2). Example: a(8)=11 because we have [8],[7,1],[6,2],[5,3],[4,4],[6,1,1],[5,2,1],[4,3,1],[4,2,2],[3,3,2], and [2,2,2,2].
Also number of partitions of 2*n into distinct parts with either all parts odd or all parts even. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Jul 03 2007
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FORMULA
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G.f.=sum(x^(k*floor(k/2))/product(1-x^j, j=1..k), k=1..infinity).
a(n) = A000700(2*n) + A000009(n), n>0. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Jul 03 2007
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EXAMPLE
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a(8)=11 because we have [4,4],[3,3,2],[3,3,1,1],[3,2,2,1],[3,2,1,1,1],[3,1,1,1,1,1],[2,2,2,2],[2,2,2,1,1],[2,2,1,1,1,1],[2,1,1,1,1,1,1], and [1,1,1,1,1,1,1,1].
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MAPLE
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g:=sum(x^(k*floor(k/2))/product(1-x^j, j=1..k), k=1..15): gser:=series(g, x=0, 65): seq(coeff(gser, x, n), n=0..60);
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CROSSREFS
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Cf. A118082, A118084.
Sequence in context: A029002 A031121 A080655 this_sequence A116470 A115649 A111795
Adjacent sequences: A118080 A118081 A118082 this_sequence A118084 A118085 A118086
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KEYWORD
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nonn
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 12 2006
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