A377526 E.g.f. satisfies A(x) = 1 + x*exp(x)*A(x)^5.

1, 1, 12, 273, 9604, 460105, 27966126, 2062219117, 178897527768, 17853102321489, 2014988044093210, 253792946798597701, 35290880970687039732, 5370055269772474994713, 887591963820839894529654, 158357028389450319651183165, 30332317748593431632078480176, 6208425034878692992471996557217

Offset: 0

Author

Seiichi Manyama, Oct 30 2024

Keywords

• nonn

Comments

In general, for k > 1, if e.g.f. satisfies A(x) = 1 + x*exp(x)*A(x)^k, then a(n) ~ sqrt(k*(1 + LambertW((k-1)^(k-1)/k^k))) * n^(n-1) / ((k-1)^(3/2) * exp(n) * LambertW((k-1)^(k-1)/k^k)^n). - Vaclav Kotesovec, Nov 11 2024

Crossrefs

Cf. A006153, A295238, A364983, A364987.

Cf. A002294.

Programs

PARI

a(n) = n!*sum(k=0, n, k^(n-k)*binomial(5*k, k)/((4*k+1)*(n-k)!));

Formula

a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(5*k,k)/( (4*k+1)*(n-k)! ) = n! * Sum_{k=0..n} k^(n-k) * A002294(k)/(n-k)!.

a(n) ~ sqrt(5*(1 + LambertW(256/3125))) * n^(n-1) / (8 * exp(n) * LambertW(256/3125)^n). - Vaclav Kotesovec, Nov 11 2024