A362693 E.g.f. satisfies A(x) = exp(x + x / A(x)).

1, 2, 0, 8, -64, 832, -13568, 269824, -6328320, 171044864, -5235245056, 178988498944, -6760886435840, 279614956503040, -12566949343002624, 609881495812702208, -31785828867471572992, 1770660964785178279936, -104990165030126886060032

Offset: 0

Author

Seiichi Manyama, May 01 2023

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..372
• Eric Weisstein's World of Mathematics, Lambert W-Function.

Crossrefs

Cf. A349562, A362694, A362734, A362735.

Cf. A362736, A362737.

Cf. A349719.

Programs

Mathematica

nmax = 20; A[_] = 1;
Do[A[x_] = Exp[x + x/A[x]] + 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*exp(-x)))))

Formula

E.g.f.: x / LambertW(x*exp(-x)) = exp( x + LambertW(x*exp(-x)) ).

a(n) = Sum_{k=0..n} (-k+1)^(n-1) * binomial(n,k) = 2^n * A349719(n).