A382001 E.g.f. A(x) satisfies A(x) = 1 + x*exp(2*x)*A(x)^6.

1, 1, 16, 462, 20672, 1261400, 97728672, 9190016416, 1016963389696, 129485497897728, 18648682990461440, 2997567408967391744, 531985786683988512768, 103321584851593487961088, 21798243872991807130685440, 4964302861788729054456729600, 1213816740632458735310221672448

Offset: 0

Author

Seiichi Manyama, Mar 12 2025

Keywords

• nonn

Comments

In general, if k>1 and e.g.f. A(x) satisfies A(x) = 1 + x*exp(2*x)*A(x)^k, then a(n) ~ sqrt(k) * sqrt(1 + LambertW(2*(k-1)^(k-1)/k^k)) * 2^n * n^(n-1) / ((k-1)^(3/2) * exp(n) * LambertW(2*(k-1)^(k-1)/k^k)^n). - Vaclav Kotesovec, Mar 22 2025

Crossrefs

Cf. A336950, A381997, A381998, A381999, A382000.

Cf. A002295, A381989.

Programs

PARI

a(n) = n!*sum(k=0, n, (2*k)^(n-k)*binomial(6*k+1, k)/((6*k+1)*(n-k)!));

Formula

a(n) = n! * Sum_{k=0..n} (2*k)^(n-k) * A002295(k)/(n-k)!.

a(n) ~ sqrt(3*(1 + LambertW(3125/23328))) * 2^(n + 1/2) * n^(n-1) / (5^(3/2) * exp(n) * LambertW(3125/23328)^n). - Vaclav Kotesovec, Mar 22 2025