A362690 E.g.f. satisfies A(x) = exp(x^2 + x * A(x)).

1, 1, 5, 28, 245, 2816, 40537, 702976, 14270153, 332102656, 8719631981, 255020847104, 8222803663549, 289815184113664, 11085650268060929, 457386463819595776, 20248713707077863953, 957435459515190345728, 48157934732749633188565

Offset: 0

Author

Seiichi Manyama, May 01 2023

Keywords

• nonn

Comments

Essentially the same as A138293.

Links

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

Crossrefs

Cf. A349562, A362691.

Cf. A125500, A138293, A362736.

Programs

PARI

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

Formula

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

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

a(n) ~ sqrt(1 + LambertW(2*exp(-2))) * 2^((n+1)/2) * n^(n-1) / (exp(n) * LambertW(2*exp(-2))^((n+1)/2)). - Vaclav Kotesovec, Nov 10 2023