A318696 Expansion of e.g.f. Product_{i>=1, j>=1} (1 + x^(i*j))^(1/(i*j)).

1, 1, 2, 10, 34, 218, 1708, 12556, 97340, 1139932, 12602584, 142757624, 1983086488, 26745019000, 402951386576, 7181178238672, 115410887636752, 2039658743085584, 42354537803172640, 815690033731561888, 17593347085888752416, 416765224159172991136, 9379433694333768563392

Offset: 0

Author

Ilya Gutkovskiy, Aug 31 2018

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..446
• 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.

Crossrefs

Cf. A000005, A107742, A168243, A280541, A288571, A318695, A318977.

Programs

Maple

seq(n!*coeff(series(mul((1+x^k)^(tau(k)/k),k=1..100),x=0,23),x,n),n=0..22); # Paolo P. Lava, Jan 09 2019

Mathematica

nmax = 22; CoefficientList[Series[Product[Product[(1 + x^(i j))^(1/(i j)), {i, 1, nmax}], {j, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Product[(1 + x^k)^(DivisorSigma[0, k]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k/d + 1) DivisorSigma[0, d], {d, Divisors[k]}] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) DivisorSigma[0, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 22}]
nmax = 22; s = 1 + x; Do[s *= Sum[Binomial[DivisorSigma[0, 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 01 2018 *)

Formula

E.g.f.: Product_{k>=1} (1 + x^k)^(tau(k)/k), where tau = number of divisors (A000005).

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