A372234 E.g.f. A(x) satisfies A(x) = exp( 2 * x * (1 + x^2 * A(x)^(1/2)) ).

1, 2, 4, 20, 160, 1112, 9424, 114788, 1453792, 19242224, 309179104, 5533486268, 102733943536, 2105041949480, 47732237414320, 1139969559931028, 28924667996076736, 792458458301707232, 22984740550326524608, 699915806697250558316

Offset: 0

Author

Seiichi Manyama, Apr 23 2024

Keywords

• nonn

Links

• Eric Weisstein's World of Mathematics, Lambert W-Function.

Crossrefs

Cf. A362392, A372156.

Programs

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(2*x-2*lambertw(-x^3*exp(x)))))

PARI

a(n) = 2*n!*sum(k=0, n\3, (k+2)^(n-2*k-1)/(k!*(n-3*k)!));

Formula

E.g.f.: A(x) = exp( 2*x - 2*LambertW(-x^3 * exp(x)) ).

a(n) = 2 * n! * Sum_{k=0..floor(n/3)} (k+2)^(n-2*k-1) / (k! * (n-3*k)!).

a(n) ~ 2*sqrt(1 + LambertW(exp(-1/3)/3)) * n^(n-1) / (3^(n + 11/2) * exp(n) * LambertW(exp(-1/3)/3)^(n+6)). - Vaclav Kotesovec, Jun 01 2024