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A191661
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Third differences of A000219.
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3
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-1, 3, 0, 9, 1, 22, 12, 48, 45, 120, 125, 290, 354, 676, 913, 1611, 2232, 3757, 5349, 8597, 12462, 19476, 28325, 43445, 63328, 95462, 139139, 207171, 301022, 443779, 642650, 939014, 1354671, 1964715, 2822084, 4066480, 5815907, 8330621, 11863720, 16902592, 23968714, 33981168, 47988828, 67722579, 95258824, 133854462, 187554809, 262483024, 366425586
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OFFSET
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0,2
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COMMENTS
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More generally, for fixed m > 0, if a(m,n) are m-fold differences of A000219, then
a(m,n) ~ A000219(n) * (2*Zeta[3]/n)^(m/3).
a(m,n) ~ Zeta(3)^(7/36 + m/3) * exp(3 * Zeta(3)^(1/3) * (n/2)^(2/3) + 1/12) / (A * sqrt(3*Pi) * 2^(11/36 - m/3) * n^(25/36 + m/3)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.
(End)
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REFERENCES
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G. Almkvist, The differences of the number of plane partitions, Manuscript, circa 1991.
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LINKS
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FORMULA
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a(n) ~ 2^(25/36) * Zeta(3)^(43/36) * exp(1/12 + 3*Zeta(3)^(1/3)*n^(2/3)/2^(2/3)) / (A * sqrt(3*Pi) * n^(61/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Oct 30 2016
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MATHEMATICA
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nmax = 50; Drop[CoefficientList[Series[(1-x)^3 * Product[1/(1-x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x], 3] (* Vaclav Kotesovec, Oct 30 2016 *)
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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