A373577 -id:A373577 - cp's OEIS frontend

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

Seiichi Manyama, Jun 10 2024

nonn

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

Cf. A118395, A190863, A373577.

Cf. A361278.

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

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

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

A373619 Expansion of e.g.f. exp(x / (1 - x^2)^(3/2)).

Original entry on oeis.org

1, 1, 1, 10, 37, 316, 2341, 21736, 237385, 2611792, 35911081, 476570656, 7654975021, 121021831360, 2196593121997, 40464132512896, 817485662059921, 17159299818547456, 382733978898335185, 8982388245979044352, 219867829220866999861, 5684505550914409716736

Offset: 0

Seiichi Manyama, Jun 11 2024

nonn

Cf. A012150, A088009, A373620.

Cf. A373577.

a(n) = n!*sum(k=0, n\2, binomial(3*n/2-2*k-1, k)/(n-2*k)!);

a(n) = n! * Sum_{k=0..floor(n/2)} binomial(3*n/2-2*k-1,k)/(n-2*k)!.
a(n) == 1 mod 9.
a(n) ~ 3^(1/5) * 5^(-1/2) * exp(3^(-1/5)*n^(1/5)/4 + 5*3^(-3/5)*n^(3/5)/2 - n) * n^(n - 1/5) * (1 - 1/(10*3^(4/5)*n^(1/5))). - Vaclav Kotesovec, Jun 11 2024

cp's OEIS Frontend

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

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