A349651 E.g.f. satisfies: A(x)^(A(x)^3) = 1 + x.

1, 1, -6, 81, -1776, 54240, -2125122, 101631558, -5739235128, 373745355984, -27572590788480, 2272763834553168, -207013811669644680, 20647997125333476912, -2238256520486195804280, 262010379635788799196360, -32939968662220720559744448

Offset: 0

Author

Seiichi Manyama, Nov 23 2021

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..331
• Eric Weisstein's World of Mathematics, Lambert W-Function.

Crossrefs

Cf. A349653, A349655, A349657.

Cf. A120980, A349650.

Programs

Mathematica

nmax = 20; A[_] = 1;
Do[A[x_] = (1 + x)^(1/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) = (-1)^(n-1)*sum(k=0, n, (3*k-1)^(k-1)*abs(stirling(n, k, 1)));

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(-sum(k=0, N, (3*k-1)^(k-1)*(-log(1+x))^k/k!)))

PARI

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

Formula

a(n) = (-1)^(n-1) * Sum_{k=0..n} (3*k-1)^(k-1) * |Stirling1(n,k)|.

E.g.f. A(x) = -Sum_{k>=0} (3*k-1)^(k-1) * (-log(1+x))^k / k!.

E.g.f.: A(x) = ( 3*log(1+x)/LambertW(3*log(1+x)) )^(1/3).

a(n) ~ -(-1)^n * n^(n-1) * exp(1/6 - n + n*exp(-1)/3) / (sqrt(3) * (exp(exp(-1)/3) - 1)^(n - 1/2)). - Vaclav Kotesovec, Nov 24 2021