A362492 E.g.f. satisfies A(x) = exp(x - x^2/2 * A(x)^2).

1, 1, 0, -8, -38, 106, 3676, 24508, -296036, -9149156, -56500064, 2211573376, 64958496472, 184823374360, -35372361487280, -971135892546224, 4364710018963216, 1034808592156017424, 25290798052846014208, -474242641154857953152, -49625273567646267051104

Offset: 0

Author

Seiichi Manyama, Apr 22 2023

Keywords

• sign

Links

• Robert Israel, Table of n, a(n) for n = 0..403
• Eric Weisstein's World of Mathematics, Lambert W-Function.

Crossrefs

Cf. A362493, A362494.

Cf. A362480.

Programs

Maple

N:= 50: # for a(0)..a(N)
egf:=  exp(x - LambertW(x^2 * exp(2*x))/2):
S:=series(egf,x,N+1):
[seq](coeff(S,x,i)*i!,i=0..N); # Robert Israel, May 22 2023

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(x^2*exp(2*x))/2)))

Formula

E.g.f.: exp(x - LambertW(x^2 * exp(2*x))/2) = sqrt( LambertW(x^2 * exp(2*x))/x^2 ).

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