A362478 E.g.f. satisfies A(x) = exp(x + x^3/3 * A(x)^3).

1, 1, 1, 3, 33, 321, 2841, 31641, 498849, 8979489, 167510961, 3427780401, 80374833441, 2089382321313, 58020408889353, 1721768971537161, 55150870311938241, 1897482353016075201, 69322763655015214689, 2676706914491568918369

Offset: 0

Author

Seiichi Manyama, Apr 21 2023

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..399
• Eric Weisstein's World of Mathematics, Lambert W-Function.

Crossrefs

Column k=2 of A362490.

Cf. A362474, A362491.

Cf. A362390.

Programs

Mathematica

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

PARI

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

Formula

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

a(n) = n! * Sum_{k=0..floor(n/3)} (1/3)^k * (3*k+1)^(n-2*k-1) / (k! * (n-3*k)!).