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A288391
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Expansion of Product_{k>=1} 1/(1 - x^k)^(sigma_3(k)).
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13
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1, 1, 10, 38, 156, 534, 2014, 6796, 23312, 76165, 247234, 780343, 2435903, 7453859, 22538336, 67130594, 197666509, 574876417, 1654464954, 4711217687, 13288453688, 37133349758, 102873771662, 282630567325, 770410193747, 2084205092693, 5598070811010
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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(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A027848(k)*a(n-k) for n > 0.
a(n) ~ exp((5*Pi)^(4/5) * Zeta(5)^(1/5) * n^(4/5) / (2^(8/5) * 3^(1/5)) - Zeta'(-3)/2) * Zeta(5)^(121/1200) / ((24*Pi)^(121/1200) * 5^(721/1200) * n^(721/1200)). - Vaclav Kotesovec, Mar 23 2018
G.f.: exp(Sum_{k>=1} sigma_4(k)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Oct 26 2018
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MAPLE
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with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
d*sigma[3](d), d=divisors(j))*a(n-j), j=1..n)/n)
end:
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MATHEMATICA
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nmax = 40; CoefficientList[Series[Product[1/(1-x^k)^DivisorSigma[3, k], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 23 2018 *)
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PROG
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(PARI) m=40; x='x+O('x^m); Vec(prod(k=1, m, 1/(1-x^k)^sigma(k, 3))) \\ G. C. Greubel, Oct 30 2018
(Magma) m:=40; R<q>:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-q^k)^DivisorSigma(3, k): k in [1..m]]) )); // G. C. Greubel, Oct 30 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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STATUS
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approved
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