OFFSET
0,3
LINKS
Winston de Greef, Table of n, a(n) for n = 0..327
Eric Weisstein's World of Mathematics, Lambert W-Function.
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
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 *)
PROG
(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)))
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Mar 01 2023
STATUS
approved