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A023877
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Expansion of Product_{k>=1} (1 - x^k)^(-k^8).
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5
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1, 1, 257, 6818, 105250, 2175491, 44988020, 796565173, 13803604854, 240522266760, 4044067171130, 65769795259820, 1051279656603367, 16517653032316394, 254354069377336990, 3847172021760617755, 57300325471166205776, 840900188345961238222, 12164188625099191500782
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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) ~ exp(5 * Pi * 2^(17/10) * n^(9/10) / (3^(21/10) * 11^(1/10)) + 315*Zeta(9)/(4*Pi^8)) / (2^(13/20) * sqrt(5) * 33^(1/20) * n^(11/20)), where Zeta(9) = A013667 = 1.0020083928260822144... . - Vaclav Kotesovec, Feb 27 2015
a(n) = (1/n)*Sum_{k=1..n} sigma_9(k)*a(n-k). - Seiichi Manyama, Mar 05 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^8, d=divisors(j)) *a(n-j), j=1..n)/n)
end:
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MATHEMATICA
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max = 18; Series[ Product[1/(1 - x^k)^k^8, {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=20; x='x+O('x^m); Vec(prod(k=1, m, 1/(1-x^k)^k^8)) \\ G. C. Greubel, Oct 31 2018
(Magma) m:=20; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-x^k)^k^8: k in [1..m]]) )); // G. C. Greubel, Oct 31 2018
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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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