A294392 E.g.f.: exp(Sum_{n>=1} A001227(n) * x^n).

1, 1, 3, 19, 97, 801, 7411, 73123, 821409, 10977697, 151612291, 2286137811, 38308830913, 669163118209, 12649211055027, 257559356068771, 5432325991339201, 121949878889492673, 2907330680764076419, 71860237654425159187, 1871308081194213959841

Offset: 0

Author

Seiichi Manyama, Oct 30 2017

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..441

Crossrefs

E.g.f.: exp(Sum_{n>=1} (Sum_{d|n and d is odd} d^k) * x^n): this sequence (k=0), A294394 (k=1), A294395 (k=2).

Cf. A001227, A206303.

Programs

Mathematica

a[n_] := a[n] = If[n == 0, 1, Sum[k*DivisorSum[k, Mod[#, 2] &]*a[n - k], {k, 1, n}]/n]; Table[n!*a[n], {n, 0, 20}] (* Vaclav Kotesovec, Sep 07 2018 *)

PARI

N=66; x='x+O('x^N); Vec(serlaplace(exp(sum(k=1, N, sumdiv(k, d, d%2)*x^k))))

Formula

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k*A001227(k)*a(n-k)/(n-k)! for n > 0.

E.g.f.: Product_{k>=1} exp(x^(2*k-1)/(1 - x^(2*k-1))). - Ilya Gutkovskiy, Nov 27 2017

Conjecture: log(a(n)/n!) ~ sqrt(n*log(n)). - Vaclav Kotesovec, Sep 07 2018