A356001 Expansion of e.g.f. -LambertW((1 - exp(3*x))/3).

0, 1, 5, 36, 379, 5461, 100476, 2250613, 59432141, 1807959042, 62262816157, 2394551966401, 101724440338494, 4730814590128615, 239057921691911861, 13042779411190737420, 764136645388807739239, 47846833035272035228849, 3188740106752561252031364

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. A048802, A356000.

Cf. A357336.

Programs

Mathematica

With[{m = 20}, Range[0, m]! * CoefficientList[Series[-ProductLog[(1 - Exp[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((1-exp(3*x))/3))))

PARI

a(n) = sum(k=1, n, 3^(n-k)*k^(k-1)*stirling(n, k, 2));

Formula

a(n) = Sum_{k=1..n} 3^(n-k) * k^(k-1) * Stirling2(n,k).

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

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