A358738 Expansion of Sum_{k>=0} k! * ( x/(1 - k*x) )^k.

1, 1, 3, 15, 103, 893, 9341, 114355, 1603155, 25318137, 444689497, 8597568671, 181430298479, 4149361409077, 102229328244837, 2699254206069387, 76038064580742091, 2276259442660623857, 72160287650141753777, 2414950992007231422007

Offset

0,3

Links

Table of n, a(n) for n=0..19.

Formula

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

a(n) ~ n! / ((1 + LambertW(1))^2 * LambertW(1)^n). - Vaclav Kotesovec, Feb 18 2023

Mathematica

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

Prog

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

Crossrefs

Cf. A001339, A006153, A080108.

Cf. A358740, A358741.

Sequence in context: A079486 A245118 A343884 * A331689 A001274 A139766

Adjacent sequences: A358735 A358736 A358737 * A358739 A358740 A358741

Keyword

nonn

Author

Seiichi Manyama, Nov 29 2022

Status

approved