A356667 Expansion of e.g.f. Sum_{k>=0} x^k / (1 - k*x^k/k!).

1, 1, 4, 12, 72, 240, 2520, 10080, 127680, 816480, 11037600, 79833600, 1514177280, 12454041600, 261655954560, 2699348652000, 62869385779200, 711374856192000, 19407798693803520, 243290200817664000, 7300765959334848000, 102980278869910041600

Offset: 0

Author

Seiichi Manyama, Aug 22 2022

Keywords

• nonn

Crossrefs

Cf. A356632, A356633, A356634.

Cf. A356668.

Programs

Mathematica

a[n_]:= n! * DivisorSum[n, 1/(# - 1)!^(n/# - 1) &]; a[0] = 1; Array[a, 22, 0] (* Amiram Eldar, Aug 22 2022 *)

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, x^k/(1-k*x^k/k!))))

PARI

a(n) = if(n==0, 1, n!*sumdiv(n, d, 1/(d-1)!^(n/d-1)));

Formula

a(n) = n! * Sum_{d|n} 1/((d-1)!^(n/d-1)) for n > 0.

a(p) = 2 * p! for prime p.