A349657 E.g.f. satisfies: A(x)^3 * log(A(x)) = 1 - exp(-x).

1, 1, -6, 80, -1751, 53402, -2088528, 99680667, -5617170700, 365003288652, -26868393676609, 2209797209486528, -200828403704351068, 19986049281174575497, -2161617056877509895386, 252467067400866652634004, -31668302130310076212791823

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. A349651, A349653, A349655.

Cf. A349585, A349656.

Programs

Mathematica

nmax = 20; A[_] = 1;
Do[A[x_] = Exp[(Exp[x]-1)/(Exp[x]*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)*stirling(n, k, 2));

PARI

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

PARI

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

Formula

a(n) = (-1)^(n-1) * Sum_{k=0..n} (3*k-1)^(k-1) * Stirling2(n,k).

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

G.f.: Sum_{k>=0} (-3*k+1)^(k-1) * x^k/Product_{j=1..k} (1 + j*x).

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