A373578 Expansion of e.g.f. exp(x * (1 + x^2)^2).

1, 1, 1, 13, 49, 241, 2401, 13021, 128353, 1346689, 10615681, 140431501, 1544877841, 17576665393, 264566466529, 3226728670621, 48376006929601, 766753039205761, 11052669865900033, 197019825098096269, 3271213100827557361, 56597110823949654001

Offset: 0

Author

Seiichi Manyama, Jun 10 2024

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A118395, A190863, A373577.

Cf. A361278.

Programs

Mathematica

nmax = 20; CoefficientList[Series[E^(x*(1 + x^2)^2), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 11 2024 *)

PARI

a(n) = n!*sum(k=0, 2*n\5, binomial(2*n-4*k, k)/(n-2*k)!);

Formula

a(n) = n! * Sum_{k=0..floor(2*n/5)} binomial(2*n-4*k,k)/(n-2*k)!.

a(n) == 1 (mod 12).

a(n) = a(n-1) + 6*(n-1)*(n-2)*a(n-3) + 5*(n-1)*(n-2)*(n-3)*(n-4)*a(n-5).

a(n) ~ 5^(n/5 - 1/2) * exp(7*5^(-11/5)*n^(1/5) + 2*5^(-3/5)*n^(3/5) - 4*n/5) * n^(4*n/5). - Vaclav Kotesovec, Jun 11 2024