A168243 Expansion of e.g.f. Product_{i>=1} (1 + x^i)^(1/i).

1, 1, 1, 5, 11, 59, 439, 2659, 13705, 160649, 2009681, 16966421, 183312931, 2078169235, 34203787591, 657685416179, 8054585463569, 104530824746129, 2595754682459425, 39767021562661669, 758079429084897211

Offset: 0

Author

Vladeta Jovovic, Nov 21 2009

Keywords

• easy
• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..446
• Vaclav Kotesovec, Graph: (a(n)/n!) / (n^(log(2) - 1)), 250000 terms
• 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. A028342.

Programs

Mathematica

nmax=20; CoefficientList[Series[Product[(1+x^k)^(1/k),{k,1,nmax}],{x,0,nmax}],x] * Range[0,nmax]! (* Vaclav Kotesovec, May 28 2015 *)
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[-(-1)^d, {d, Divisors[k]}]*a[n-k], {k, 1, n}]/n]; Table[n!*a[n], {n, 0, 20}] (* Vaclav Kotesovec, Sep 07 2018 *)

Formula

E.g.f.: exp(Sum_{n>=1} A048272(n)*x^n/n).

Conjecture: log(a(n)/n!) ~ (log(2) - 1) * log(n). - Vaclav Kotesovec, Sep 10 2018