A376107 Expansion of e.g.f. LambertW(x / (1 - 3*x)).

0, 1, 4, 27, 260, 3265, 50634, 935263, 20053816, 489677697, 13416375950, 407609962111, 13600700469828, 494442286466401, 19452778285314178, 823489845351967935, 37323572563440199664, 1803303384581598518785, 92523649833821902792086

Offset: 0

Author

Seiichi Manyama, Sep 10 2024

Keywords

• nonn

Links

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

Crossrefs

Cf. A060356, A376106.

Cf. A376101.

Programs

Mathematica

nmax=20; CoefficientList[InverseSeries[Series[x / (3*x + E^(-x)), {x, 0, nmax}], x], x] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 20 2024 *)

PARI

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

PARI

a(n) = n!*sum(k=1, n, 3^(n-k)*(-k)^(k-1)*binomial(n-1, k-1)/k!);

Formula

E.g.f. A(x) satisfies A(x) = x * (3*A(x) + exp(-A(x))).

E.g.f.: Series_Reversion( x / (3*x + exp(-x)) ).

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