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Search: id:A118084
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| A118084 |
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Number of partitions of n such that largest part k occurs at most floor(k/2) times. |
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3
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| 0, 1, 2, 3, 5, 7, 11, 16, 23, 33, 46, 63, 86, 116, 153, 203, 265, 345, 444, 571, 727, 925, 1166, 1468, 1836, 2293, 2845, 3525, 4345, 5347, 6550, 8011, 9758, 11867, 14380, 17399, 20984, 25269, 30341, 36376, 43500, 51943, 61877, 73608, 87373, 103571
(list; graph; listen)
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OFFSET
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1,3
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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 most floor(k/2). Example: a(8)=16 because we have [7,1],[6,1,1],[5,2,1],[4,3,1],[5,1,1,1],[4,2,1,1],[3,3,1,1],[3,2,2,1],[2,2,2,2],[4,1,1,1,1],[3,2,1,1,1],[2,2,2,1,1],[3,1,1,1,1,1],[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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FORMULA
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G.f.=sum(x^k*(1-x^(k(floor(k/2))))/product(1-x^j, j=1..k), k=1..infinity).
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EXAMPLE
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a(8)=16 because we have [8],[7,1],[6,2],[6,1,1],[5,3],[5,2,1],[5,1,1,1],[4,4],[4,3,1],[4,2,2],[4,2,1,1],[4,1,1,1,1],[3,2,2,1],[3,2,1,1,1],[3,1,1,1,1,1], and [2,1,1,1,1,1,1].
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MAPLE
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g:=sum(x^k*(1-x^(k*(floor(k/2))))/product(1-x^j, j=1..k), k=1..85): gser:=series(g, x=0, 55): seq(coeff(gser, x, n), n=1..50);
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CROSSREFS
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Cf. A118082, A118083.
Sequence in context: A052955 A022480 A024791 this_sequence A117590 A023435 A091501
Adjacent sequences: A118081 A118082 A118083 this_sequence A118085 A118086 A118087
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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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