A371038 E.g.f. satisfies A(x) = exp(x^2*A(x)) / (1-x).

1, 1, 4, 18, 132, 1140, 12720, 164640, 2514960, 43500240, 850076640, 18418609440, 439831909440, 11457415569600, 323707663319040, 9854548934630400, 321709145793235200, 11209975693710393600, 415330670608805952000, 16303720885477254028800

Offset: 0

Author

Seiichi Manyama, Mar 09 2024

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 0..399
• Eric Weisstein's World of Mathematics, Lambert W-Function.

Crossrefs

Cf. A352410, A371039.

Programs

Maple

E:=  LambertW( -x^2/(1-x) ) / (-x^2):
S:= series(E,x,43):
seq(coeff(S,x,i)*i!,i=0..40); # Robert Israel, Mar 09 2025

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(lambertw(-x^2/(1-x))/(-x^2)))

PARI

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

Formula

E.g.f.: LambertW( -x^2/(1-x) ) / (-x^2).

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

a(n) ~ exp(2) * sqrt(1 + 4*exp(1) - sqrt(1 + 4*exp(1))) * 2^(n + 3/2) * n^(n-1) / ((1 + 2*exp(1) - sqrt(1 + 4*exp(1)))*(-1 + sqrt(1 + 4*exp(1)))^(n+1)). - Vaclav Kotesovec, Mar 12 2024