A356000 Expansion of e.g.f. -LambertW((1 - exp(2*x))/2).

0, 1, 4, 25, 236, 3061, 50670, 1020881, 24245576, 663290281, 20541118266, 710366714773, 27135242829436, 1134708855427629, 51556563327940390, 2529164265815033241, 133229047747850647312, 7500633471737652798673, 449445732625670948076530

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, A356001.

Cf. A357335.

Programs

Mathematica

With[{m = 20}, Range[0, m]! * CoefficientList[Series[-ProductLog[(1 - Exp[2*x])/2], {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(2*x))/2))))

PARI

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

Formula

a(n) = Sum_{k=1..n} 2^(n-k) * k^(k-1) * Stirling2(n,k).

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

E.g.f.: Series_Reversion( (log(1 + 2 * x * exp(-x)))/2 ). - Seiichi Manyama, Sep 11 2024