A361066 E.g.f. satisfies A(x) = exp( (x/(1-x)) * A(x)^3 ).

1, 1, 9, 148, 3673, 123276, 5234599, 269262022, 16279709793, 1131627199816, 88926737901031, 7796168316687906, 754414052156289265, 79872584117422215484, 9184299004593618881655, 1139822558262829096519726, 151857077047173825979147969

Offset: 0

Author

Seiichi Manyama, Mar 01 2023

Keywords

• nonn

Links

• Winston de Greef, Table of n, a(n) for n = 0..327
• Eric Weisstein's World of Mathematics, Lambert W-Function.

Crossrefs

Cf. A052868, A361065, A361067, A361068, A361069.

Programs

Mathematica

nmax = 20; A[_] = 1;
Do[A[x_] = Exp[(x/(1 - x))*A[x]^3] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)

PARI

a(n) = n!*sum(k=0, n, (3*k+1)^(k-1)*binomial(n-1, n-k)/k!);

PARI

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

PARI

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

Formula

a(n) = n! * Sum_{k=0..n} (3*k+1)^(k-1) * binomial(n-1,n-k)/k!.

E.g.f.: exp( -LambertW(-3*x/(1-x))/3 ).

E.g.f.: ( -(1-x)/(3*x) * LambertW(-3*x/(1-x)) )^(1/3).

a(n) ~ (1 + 3*exp(1))^(n + 1/2) * n^(n-1) / (3^(3/2) * exp(n + 1/6)). - Vaclav Kotesovec, Mar 02 2023