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A326779
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E.g.f.: Product_{k>=1} 1/(1 - x^(4*k-1)/(4*k-1)).
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4
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1, 0, 0, 2, 0, 0, 80, 720, 0, 13440, 172800, 3628800, 5913600, 98841600, 4420915200, 92559667200, 110702592000, 6012444672000, 234205087334400, 6616915329024000, 13708373852160000, 771938716483584000, 40374130262409216000, 1172555787961958400000
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OFFSET
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0,4
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COMMENTS
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In the article by Lehmer, Theorem 7, p. 387, case b <> 0 and b <> 1, correct formula is W_n(S_a,b) ~ a^(-1/a) * exp(-gamma/a) * (Gamma((b-1)/a) / (Gamma(b/a) * Gamma(1/a))) * n^(1/a - 1), where gamma is the Euler-Mascheroni constant (A001620) and Gamma() is the Gamma function.
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LINKS
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Vaclav Kotesovec, Table of n, a(n) for n = 0..449
Vaclav Kotesovec, Graph - the asymptotic ratio (50000 terms)
D. H. Lehmer, On reciprocally weighted partitions, Acta Arithmetica XXI (1972), 379-388 (Theorem 7 needs a correction).
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FORMULA
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a(n) ~ exp(-gamma/4) * n! / (2 * sqrt(Pi) * n^(3/4)), where gamma is the Euler-Mascheroni constant A001620.
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MATHEMATICA
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nmax = 25; CoefficientList[Series[1/Product[(1-x^(4*k-1)/(4*k-1)), {k, 1, Floor[nmax/4] + 1}], {x, 0, nmax}], x] * Range[0, nmax]!
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CROSSREFS
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Cf. A007841, A294506, A309319, A326755, A326756, A326780.
Sequence in context: A013416 A156433 A169771 * A293140 A008551 A183896
Adjacent sequences: A326776 A326777 A326778 * A326780 A326781 A326782
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KEYWORD
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nonn
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AUTHOR
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Vaclav Kotesovec, Jul 24 2019
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STATUS
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approved
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