A356925 E.g.f. satisfies A(x) = 1/(1 - x)^(exp(x) * A(x)).

1, 1, 6, 51, 614, 9655, 188209, 4389532, 119363488, 3711190881, 129932611723, 5060364817200, 217054300138136, 10168837756846145, 516709033266165479, 28306732060349788908, 1663231006737554997168, 104344911495734048046929

Offset

0,3

Links

Table of n, a(n) for n=0..17.

Eric Weisstein's World of Mathematics, Lambert W-Function.

Formula

E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (-exp(x) * log(1-x))^k / k!.

E.g.f.: A(x) = exp( -LambertW(exp(x) * log(1-x)) ).

E.g.f.: A(x) = LambertW(exp(x) * log(1-x))/(exp(x) * log(1-x)).

a(n) ~ sqrt(1 + exp(1+r)/(1-r)) * n^(n-1) / (r^(n - 1/2) * exp(n-1)), where r = 0.249272970940807862774650581662931601739615720771408527... is the root of the equation exp(1+r) * log(1-r) = -1. - Vaclav Kotesovec, Nov 14 2022

Mathematica

nmax = 20; CoefficientList[Series[LambertW[E^x * Log[1-x]]/(E^x * Log[1-x]), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 14 2022 *)

Prog

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

Crossrefs

Cf. A191365, A356926.

Cf. A356752, A356753.

Cf. A000272, A002104, A356927.

Sequence in context: A293128 A304185 A215003 * A134525 A125865 A210810

Adjacent sequences: A356922 A356923 A356924 * A356926 A356927 A356928

Keyword

nonn

Author

Seiichi Manyama, Sep 04 2022

Status

approved