A356668 Expansion of e.g.f. Sum_{k>=0} x^k / (k! - k*x^k).

1, 1, 3, 7, 37, 121, 1141, 5041, 60761, 378001, 5444461, 39916801, 729041545, 6227020801, 130767460825, 1321314894901, 31388220966961, 355687428096001, 9636906872926477, 121645100408832001, 3649432697160095561, 51223991519836175041, 1686001091666419279753

Offset: 0

Author

Seiichi Manyama, Aug 22 2022

Keywords

• nonn

Crossrefs

Cf. A356029, A356328, A356608.

Cf. A038507, A327578, A356667.

Programs

Mathematica

a[n_]:= n! * DivisorSum[n, 1/(# * (# - 1)!^(n/#)) &]; a[0] = 1; Array[a, 23, 0] (* Amiram Eldar, Aug 22 2022 *)

PARI

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

PARI

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

Formula

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

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

a(p) = 1 + p! for prime p.