A373620 Expansion of e.g.f. exp(x / (1 - x^2)^2).

1, 1, 1, 13, 49, 481, 3841, 38221, 464353, 5368609, 82042561, 1151767981, 20242097041, 342921513793, 6705416722369, 133590317946541, 2880298682358721, 65597610230669761, 1556262483879791233, 39569880403136366029, 1030778206965403668721

Offset: 0

Author

Seiichi Manyama, Jun 11 2024

Keywords

• nonn

Crossrefs

Cf. A012150, A088009, A373619.

Cf. A082579, A373578.

Programs

Maple

A373620 := proc(n)
    add(binomial(2*n-3*k-1,k)/(n-2*k)!,k=0..floor(n/2)) ;
    %*n! ;
end proc:
seq(A373620(n),n=0..80) ; # R. J. Mathar, Jun 11 2024

PARI

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

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} binomial(2*n-3*k-1,k)/(n-2*k)!.

a(n) == 1 mod 12.

a(n) ~ 2^(-1/6) * 3^(-1/2) * exp(1/48 + 2^(-5/3)*n^(1/3) + 3*2^(-4/3)*n^(2/3) - n) * n^(n - 1/6). - Vaclav Kotesovec, Jun 11 2024

D-finite with recurrence a(n) -a(n-1) -3*(n-1)*(n-2)*a(n-2) -3*(n-1)*(n-2)*a(n-3) +3*(n-1)*(n-2)*(n-3)*(n-4)*a(n-4) -(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(n-6)*a(n-6)=0. - R. J. Mathar, Jun 11 2024