OFFSET
0,2
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..329
Eric Weisstein's World of Mathematics, Lambert W-Function.
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
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 *)
PROG
(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)))
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Mar 03 2023
STATUS
approved