A362301 a(n) = n! * Sum_{k=0..floor(n/3)} n^k * binomial(n-2*k,k)/(n-2*k)!.

1, 1, 1, 19, 97, 301, 13681, 124951, 647809, 46543897, 612367201, 4447574011, 436897375201, 7505523945349, 70104150466897, 8735878156045951, 185209511009456641, 2114594302777738801, 319284313084581674689, 8053189772356178472547

Offset: 0

Author

Seiichi Manyama, Apr 15 2023

Keywords

• nonn

Links

• Winston de Greef, Table of n, a(n) for n = 0..402
• Eric Weisstein's World of Mathematics, Lambert W-Function.

Crossrefs

Cf. A362043.

Programs

PARI

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((-lambertw(-3*x^3)/3)^(1/3))/(1+lambertw(-3*x^3))))

Formula

a(n) = A362043(n,6*n).

a(n) = n! * [x^n] exp(x + n*x^3).

E.g.f.: exp( ( -LambertW(-3*x^3)/3 )^(1/3) ) / (1 + LambertW(-3*x^3)).

a(n) ~ (1 + 2*cos(2*Pi*mod(n,3)/3 - 3^(1/6)/2)*exp(-3^(2/3)/2)) * 3^(n/3 - 1/2) * n^n / exp(2*n/3 - 1/3^(1/3)). - Vaclav Kotesovec, Apr 18 2023