A376105 Expansion of e.g.f. -LambertW(-3*x / (1 + x))/3.

0, 1, 4, 51, 948, 24465, 802098, 31975335, 1501332696, 81158916897, 4964709729510, 339064260058359, 25573087919369268, 2111171271497336529, 189350082996145020810, 18334276660240212722535, 1906166280260835065912112, 211792366386481088490433857

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

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

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

a(n) ~ (3-exp(-1))^(n + 1/2) * n^(n-1) / 3^(3/2). - Vaclav Kotesovec, Sep 11 2024

Prog

(PARI) my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-lambertw(-3*x/(1+x))/3)))
(PARI) a(n) = n!*sum(k=1, n, (-1)^(n-k)*(3*k)^(k-1)*binomial(n-1, k-1)/k!);

Crossrefs

Cf. A060356, A376104.

Cf. A376099.

Sequence in context: A328931 A343572 A336608 * A349653 A235325 A230401

Adjacent sequences: A376102 A376103 A376104 * A376106 A376107 A376108

Keyword

nonn

Author

Seiichi Manyama, Sep 10 2024

Status

approved