A373620 - OEIS

%I #15 Jun 11 2024 08:17:41

%S 1,1,1,13,49,481,3841,38221,464353,5368609,82042561,1151767981,

%T 20242097041,342921513793,6705416722369,133590317946541,

%U 2880298682358721,65597610230669761,1556262483879791233,39569880403136366029,1030778206965403668721

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

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

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

%F 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

%F 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

%p A373620 := proc(n)

%p     add(binomial(2*n-3*k-1,k)/(n-2*k)!,k=0..floor(n/2)) ;

%p     %*n! ;

%p end proc:

%p seq(A373620(n),n=0..80) ; # _R. J. Mathar_, Jun 11 2024

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

%Y Cf. A012150, A088009, A373619.

%Y Cf. A082579, A373578.

%K nonn

%O 0,4

%A _Seiichi Manyama_, Jun 11 2024