A294394 E.g.f.: exp(Sum_{n>=1} A000593(n) * x^n).

1, 1, 3, 31, 145, 1641, 17731, 194503, 2676801, 40644145, 667689571, 11514903951, 227665389073, 4578990563161, 100913115588195, 2372334731747191, 57930324367791361, 1509398686720812513, 41341036374519788611, 1184009909077133031295

Offset: 0

Author

Seiichi Manyama, Oct 30 2017

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..434

Crossrefs

E.g.f.: exp(Sum_{n>=1} (Sum_{d|n and d is odd} d^k) * x^n): A294392 (k=0), this sequence (k=1), A294395 (k=2).

Cf. A000009, A000593, A301799, A301800.

Programs

Mathematica

a[n_] := a[n] = If[n == 0, 1, Sum[k*Sum[-(-1)^d*k/d, {d, Divisors[k]}]*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*(d%2))*x^k))))

Formula

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k*A000593(k)*a(n-k)/(n-k)! for n > 0.

a(n) ~ Pi^(1/3) * exp((3*Pi)^(2/3) * n^(2/3) / 2^(4/3) - 1/24 - n) * n^(n - 1/6) / (2^(1/6) * 3^(2/3)). - Vaclav Kotesovec, Sep 07 2018