A381998 E.g.f. A(x) satisfies A(x) = 1 + x*exp(2*x)*A(x)^2.

1, 1, 8, 90, 1472, 31920, 865152, 28197904, 1075122176, 46976064768, 2315080816640, 127068467480064, 7688296957870080, 508450036968779776, 36490818871396499456, 2824787199565881477120, 234622076533699738861568, 20813348299168251651883008, 1964063064959266899440959488

Offset: 0

Author

Seiichi Manyama, Mar 12 2025

Keywords

• nonn

Crossrefs

Cf. A336950, A381997, A381999, A382000, A382001.

Cf. A000108, A295238, A379885.

Programs

PARI

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

Formula

a(n) = n! * Sum_{k=0..n} (2*k)^(n-k) * A000108(k)/(n-k)!.

From Vaclav Kotesovec, Mar 22 2025: (Start)

E.g.f.: 2/(1 + sqrt(1 - 4*exp(2*x)*x)).

a(n) ~ sqrt(1 + LambertW(1/2)) * 2^(n + 1/2) * n^(n-1) / (exp(n) * LambertW(1/2)^n). (End)