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A318784 Expansion of Product_{k>=1} 1/(1 - x^k)^(sigma_1(k)-k), where sigma_1(k) = sum of divisors of k (A000203). 5
1, 0, 1, 1, 4, 2, 11, 6, 25, 20, 56, 44, 139, 107, 283, 266, 619, 567, 1317, 1242, 2680, 2705, 5403, 5539, 10947, 11339, 21291, 23013, 41494, 45213, 79991, 88312, 151546, 170908, 284901, 324421, 532505, 611227, 981002, 1142000, 1797451, 2105773, 3268765, 3855050, 5889704, 7004451 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Convolution of A061256 and A073592.

Euler transform of A001065.

LINKS

Table of n, a(n) for n=0..45.

N. J. A. Sloane, Transforms

FORMULA

G.f.: Product_{k>=1} 1/(1 - x^k)^A001065(k).

G.f.: exp(Sum_{k>=1} sigma_2(k)*x^(2*k)/(k*(1 - x^k))), where sigma_2(k) = sum of squares of divisors of k (A001157).

a(n) ~ exp(3^(2/3) * c^(1/3) * n^(2/3)/2 - Pi^2 * n^(1/3) / (4 * 3^(2/3) * c^(1/3)) - Pi^4/(288*c) - 1/8) * A^(3/2) * c^(1/8) / (3^(5/8) * (2*Pi)^(11/24) * n^(5/8)), where c = (Pi^2 - 6)*Zeta(3) and A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Sep 03 2018

MAPLE

with(numtheory):

a:= proc(n) option remember; `if`(n=0, 1, add(add(d*

      (sigma(d)-d), d=divisors(j))*a(n-j), j=1..n)/n)

    end:

seq(a(n), n=0..50);  # Alois P. Heinz, Sep 03 2018

MATHEMATICA

nmax = 45; CoefficientList[Series[Product[1/(1 - x^k)^(DivisorSigma[1, k] - k), {k, 1, nmax}], {x, 0, nmax}], x]

nmax = 45; CoefficientList[Series[Exp[Sum[DivisorSigma[2, k] x^(2 k)/(k (1 - x^k)), {k, 1, nmax}]], {x, 0, nmax}], x]

a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (DivisorSigma[1, d] - d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 45}]

CROSSREFS

Cf. A000203, A001065, A001157, A061256, A073592, A318783.

Sequence in context: A191728 A191434 A163918 * A154699 A213568 A185878

Adjacent sequences:  A318781 A318782 A318783 * A318785 A318786 A318787

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Sep 03 2018

STATUS

approved

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Last modified April 5 23:20 EDT 2020. Contains 333260 sequences. (Running on oeis4.)