A305127 Expansion of e.g.f. Product_{k>=1} 1/(1 - x^k)^(sigma(k)/k), where sigma(k) is the sum of the divisors of k.

1, 1, 5, 23, 179, 1279, 13699, 135085, 1764377, 22527521, 344625461, 5283739471, 94562354875, 1685808248383, 33947023942259, 694786150879829, 15613612524749489, 357353282848083265, 8880505496901812197, 224851013929747732231, 6106205671049245677251, 169523515381173773551871

Offset: 0

Author

Ilya Gutkovskiy, May 26 2018

Keywords

• nonn

Comments

a(n)/n! is the Euler transform of [1, 3/2, 4/3, 7/4, 6/5, ... = sums of reciprocals of divisors of 1, 2, 3, 4, 5, ...].

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..438
• Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, La Matematica (2024). Preprint available as arXiv:2303.02240 [math.CO], 2023.
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000203, A007429, A017665, A017666, A028342, A053529, A061256, A072169, A318769, A318814.

Programs

Maple

with(numtheory): a := proc(n) option remember; `if`(n = 0, 1, add(add(sigma(d), d = divisors(j))*a(n-j), j = 1..n)/n) end proc; seq(n!*a(n), n = 0..20); # Vaclav Kotesovec, Sep 04 2018

Mathematica

nmax = 21; CoefficientList[Series[Product[1/(1 - x^k)^(DivisorSigma[1, k]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Exp[Sum[Sum[x^(j k)/(j k (1 - x^(j k))), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d DivisorSigma[-1, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 21}]
nmax = 21; s = 1 - x; Do[s *= Sum[Binomial[DivisorSigma[1, k]/k, j]*(-1)^j*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]];, {k, 2, nmax}]; CoefficientList[Series[1/s, {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 03 2018 *)

Formula

E.g.f.: Product_{k>=1} 1/(1 - x^k)^(A017665(k)/A017666(k)).

E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} x^(j*k)/(j*k*(1 - x^(j*k)))).

log(a(n)/n!) ~ sqrt(n) * Pi^2 / 3. - Vaclav Kotesovec, Sep 04 2018