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Search: id:A109085
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| A109085 |
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G.f. A(x) satisfies: A(x) = G000041(x*A(x)) where G000041(x) is the g.f. of the partition numbers A000041. |
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2
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| 1, 1, 3, 10, 38, 153, 646, 2816, 12585, 57343, 265401, 1244256, 5896512, 28200365, 135935424, 659754072, 3221354296, 15812501100, 77985955410, 386254209762, 1920391362054, 9580985321554, 47951223856445, 240680464689600
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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a(n)=Sum[Product(1 + n/h(v)^2)]/(n+1), where the product is over all boxes v in the Ferrers diagram of a partition L of n, h(v) is the hook length of v and the summation is over all partitions L of n. Example: a(3)=10 because for the partitions L=(3), (2,1), (1,1,1) of n=3 the hook length multi-sets are {3,2,1}, {3,1,1},{3,2,1}, respectively, the products are (1+3/9)(1+3/4)(1+3/1)=28/3, (1+3/9)(1+3/1)(1+3/1)=64/3, (1+3/9)(1+3/4)(1+3/1)=28/3 and now a(3)=(1/4)(28+64+28)/3=10. - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 15 2008
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REFERENCES
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Guo-Niu Han, An explicit expansion formula for the powers of the Euler product in terms of partition hook lengths, arXiv:0804.1849v3 [math.CO] 9 May 2008 (p. 5).
Guo-Niu Han, The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications, arXiv:0805.1398v1 [math.CO] 9 May 2008 (p. 5).
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FORMULA
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G.f.: A(x) = (1/x)*series_reversion(x*eta(x)). G.f.: A(x) = 1/G109084(x) where G109084(x) is g.f. of A109084.
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PROGRAM
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(PARI) a(n)=polcoeff(1/x*serreverse(x*eta(x+x*O(x^n))), n)
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CROSSREFS
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Cf. A109084, A000041.
Sequence in context: A151059 A151060 A151061 this_sequence A001002 A151062 A000902
Adjacent sequences: A109082 A109083 A109084 this_sequence A109086 A109087 A109088
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Jun 18 2005
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