A361182 E.g.f. satisfies A(x) = exp( 3*x*A(x) ) / (1-x).

1, 4, 41, 735, 19293, 672573, 29342241, 1540097541, 94579646553, 6656561754345, 528414534842949, 46716837535074897, 4552821617337191637, 484953672676323320109, 56056228305888242732841, 6988787950179969557086797, 934866118278080385555647025

Offset: 0

Author

Seiichi Manyama, Mar 03 2023

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..329
• Eric Weisstein's World of Mathematics, Lambert W-Function.

Crossrefs

Cf. A352410, A352448.

Cf. A361066.

Programs

Mathematica

nmax = 20; A[_] = 1;
Do[A[x_] = Exp[3*x*A[x]]/(1 - x) + 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*(k+1)^(k-1)*binomial(n, k)/k!);

PARI

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

Formula

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

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

a(n) ~ (1 + 3*exp(1))^(n + 3/2) * n^(n-1) / (3^(3/2) * exp(n + 1/2)). - Vaclav Kotesovec, Mar 03 2023