A295794 Expansion of e.g.f. Product_{k>=1} exp(x^k/(1 + x^k)).

1, 1, 1, 13, 25, 241, 2761, 14701, 153553, 1903105, 27877681, 263555821, 4788201001, 65083782193, 1040877257785, 24098794612621, 373918687272481, 7393663746307201, 164894196647876833, 3504497611085823565, 81863829346282866361, 2257321249626793901041, 49755091945025205954601

Offset: 0

Author

Ilya Gutkovskiy, Nov 27 2017

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..440

Crossrefs

Cf. A028342, A048272, A168243, A206303, A294363, A294392, A295739.

Programs

Maple

a:=series(mul(exp(x^k/(1+x^k)),k=1..100),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 27 2019

Mathematica

nmax = 22; CoefficientList[Series[Product[Exp[x^k/(1 + x^k)], {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[x D[Log[Product[(1 + x^k)^(1/k), {k, 1, nmax}]], x]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, (n - 1)! Sum[-k Sum[(-1)^d, {d, Divisors[k]}] a[n - k]/(n - k)!, {k, 1, n}]]; Table[a[n], {n, 0, 22}]

Formula

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

E.g.f.: exp(x*f'(x)), where f(x) = log(Product_{k>=1} (1 + x^k)^(1/k)).

a(n) ~ exp(2*sqrt(n*log(2)) - 1/4 - n) * n^(n - 1/4) * log(2)^(1/4) / sqrt(2). - Vaclav Kotesovec, Sep 07 2018