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A023872
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Expansion of Product_{k>=1} (1 - x^k)^(-k^3).
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20
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1, 1, 9, 36, 136, 477, 1703, 5746, 19099, 61622, 195366, 607069, 1856516, 5586870, 16579850, 48549116, 140438966, 401592524, 1136121837, 3181700219, 8825733603, 24261363403, 66124058839, 178757752892, 479513547399, 1276792213203, 3375707760306, 8864712158225
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) ~ (3*Zeta(5))^(59/600) * exp(5 * n^(4/5) * (3*Zeta(5))^(1/5) / 2^(7/5) + Zeta'(-3)) / (2^(41/200) * n^(359/600) * sqrt(5*Pi)), where Zeta(5) = A013663 = 1.036927755143369926..., Zeta'(-3) = ((gamma + log(2*Pi) - 11/6)/30 - 3*Zeta'(4)/Pi^4)/4 = 0.00537857635777430114441697421... . - Vaclav Kotesovec, Feb 27 2015
a(n) = (1/n)*Sum_{k=1..n} sigma_4(k)*a(n-k). - Seiichi Manyama, Mar 04 2017
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MAPLE
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with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1,
add(add(d*d^3, d=divisors(j)) *a(n-j), j=1..n)/n)
end:
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MATHEMATICA
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max = 27; Series[ Product[ 1/(1-x^k)^k^3, {k, 1, max}], {x, 0, max}] // CoefficientList[#, x]& (* Jean-François Alcover, Mar 05 2013 *)
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PROG
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(PARI) m=30; x='x+O('x^m); Vec(prod(k=1, m, 1/(1-x^k)^k^3)) \\ G. C. Greubel, Oct 30 2018
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-x^k)^k^3: k in [1..m]]) )); // G. C. Greubel, Oct 30 2018
(SageMath) # uses[EulerTransform from A166861]
b = EulerTransform(lambda n: n^3)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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