A361069 E.g.f. satisfies A(x) = exp( x/((1-x) * A(x)^3) ).

1, 1, -3, 40, -719, 18396, -598157, 23713726, -1108701519, 59735988424, -3644505746549, 248358786667674, -18697767289462967, 1541202721786228060, -138046868771541971373, 13351368704222195975206, -1386710317839048140282783, 153939247458296219191539984

Offset: 0

Author

Seiichi Manyama, Mar 01 2023

Keywords

• sign

Links

• Winston de Greef, Table of n, a(n) for n = 0..338
• Eric Weisstein's World of Mathematics, Lambert W-Function.

Crossrefs

Cf. A052868, A361065, A361066, A361067, A361068.

Programs

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 *)

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/((1-x)/(3*x)*lambertw(3*x/(1-x)))^(1/3)))

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 / ( (1-x)/(3*x) * LambertW(3*x/(1-x)) )^(1/3).

a(n) ~ (-1)^(n+1) * 3^(-3/2) * exp(-1/3) * (3 - exp(-1))^(n + 1/2) * n^(n-1). - Vaclav Kotesovec, Apr 22 2024