A318769 Expansion of e.g.f. Product_{k>=1} (1 + x^k)^(sigma(k)/k), where sigma(k) is the sum of the divisors of k.

1, 1, 3, 17, 83, 639, 5749, 53227, 561273, 7216577, 94292531, 1352253561, 21657812923, 359338829407, 6460367397093, 126124578755939, 2527688612931569, 54137820027005697, 1236730462664172643, 29137619131277727457, 725282418459957414051, 18981526480933601454911

Offset

0,3

Comments

a(n)/n! is the weigh 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..440

Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, arXiv:2303.02240 [math.CO], 2023.

N. J. A. Sloane, Transforms

Formula

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

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

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

a(n)/n! ~ c * exp(sqrt(n/2)*Pi^2/3) / n^(3/4 + log(2)/4), where c = 0.15653645678497413538057076667218805302154965061194080137... - Vaclav Kotesovec, Sep 05 2018

Maple

with(numtheory): a := proc(n) option remember; `if`(n = 0, 1, add(add(-(-1)^(j/d)*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 + x^k)^(DivisorSigma[1, k]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k + 1) 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[(-1)^(k/d + 1) 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]*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}]; Take[CoefficientList[s, x], nmax + 1] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 03 2018 *)

Crossrefs

Cf. A000203, A017665, A017666, A168243, A192065, A288417, A305127, A318696, A318814.

Sequence in context: A217623 A083217 A225307 * A344228 A216681 A350456

Adjacent sequences: A318766 A318767 A318768 * A318770 A318771 A318772

Keyword

nonn

Author

Ilya Gutkovskiy, Sep 03 2018

Status

approved