A358740 Expansion of Sum_{k>=0} k! * ( k * x/(1 - k*x) )^k.

1, 1, 9, 195, 7699, 482309, 43994741, 5508667927, 906931827831, 189998213001033, 49359340639141993, 15573690455085072011, 5866304418414451865723, 2600416934781350100016717, 1340037064604153376788884701, 794358527033920600533985973631

Offset: 0

Author

Seiichi Manyama, Nov 29 2022

Keywords

• nonn

Crossrefs

Cf. A358738, A358741.

Cf. A195242.

Programs

Mathematica

nmax = 20; CoefficientList[1 + Series[Sum[k! * (k * x/(1 - k*x))^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 18 2023 *)

PARI

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, k!*(k*x/(1-k*x))^k))

PARI

a(n) = if(n==0, 1, sum(k=1, n, k!*k^n*binomial(n-1, k-1)));

Formula

a(n) = Sum_{k=1..n} k! * k^n * binomial(n-1,k-1) for n > 0.

a(n) ~ exp(exp(-1)) * n! * n^n. - Vaclav Kotesovec, Feb 18 2023