A357322 Expansion of e.g.f. -LambertW(log(1 - 3*x)/3).

0, 1, 5, 45, 586, 10024, 213084, 5428072, 161475320, 5501761488, 211466328400, 9057714349672, 428022643010544, 22127292215218072, 1242503403120434168, 75319473068729478360, 4902798528238919060224, 341102498012848479889408

Offset: 0

Author

Seiichi Manyama, Sep 24 2022

Keywords

• nonn

Links

• Eric Weisstein's World of Mathematics, Lambert W-Function.
• Index entries for reversions of series

Crossrefs

Cf. A052807, A357321.

Cf. A357334.

Programs

Mathematica

With[{m = 20}, Range[0, m]! * CoefficientList[Series[-ProductLog[Log[1 - 3*x]/3], {x, 0, m}], x]] (* Amiram Eldar, Sep 24 2022 *)

PARI

my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-lambertw(log(1-3*x)/3))))

PARI

a(n) = sum(k=1, n, 3^(n-k)*k^(k-1)*abs(stirling(n, k, 1)));

Formula

a(n) = Sum_{k=1..n} 3^(n-k) * k^(k-1) * |Stirling1(n,k)|.

a(n) ~ 3^(n - 1/2) * n^(n-1) / ((-1 + exp(3*exp(-1)))^(n - 1/2) * exp(n - 1/2 - 3*n*exp(-1))). - Vaclav Kotesovec, Oct 04 2022

E.g.f.: Series_Reversion( (1 - exp(-3 * x * exp(-x)))/3 ). - Seiichi Manyama, Sep 11 2024