login
a(n) = ((1+n)*floor(1+n/2))*(n!/floor(1+n/2)!)^2.
1

%I #10 Oct 01 2023 18:07:54

%S 1,2,6,72,240,7200,25200,1411200,5080320,457228800,1676505600,

%T 221298739200,821966745600,149597947699200,560992303872000,

%U 134638152929280000,508633022177280000,155641704786247680000,591438478187741184000,224746621711341649920000

%N a(n) = ((1+n)*floor(1+n/2))*(n!/floor(1+n/2)!)^2.

%F a(n) = n!*A212303(n+1).

%F a(n) = (n+1)!*A057977(n).

%F a(n) = A093005(n+1)*A262033(n)^2.

%F a(n) = A093005(n+1)*A329964(n).

%F a(2*n) = A052510(n) (n >= 0).

%F a(2*n+1) = A123072(n+1) (n >= 0).

%F a(n) = n! [x^n] (1 - sqrt(1 - 4*x^2) - 4*x^2*(1 - x - sqrt(1 - 4*x^2)))/(2*x^2*(1 - 4*x^2)^(3/2)).

%p A329965 := n -> ((1+n)*floor(1+n/2))*(n!/floor(1+n/2)!)^2:

%p seq(A329965(n), n=0..19);

%t ser := Series[(1 - Sqrt[1 - 4 x^2] - 4 x^2 (1 - x - Sqrt[1 - 4 x^2]))/(2 x^2 (1 - 4 x^2)^(3/2)), {x, 0, 22}]; Table[n! Coefficient[ser, x, n], {n, 0, 20}]

%t Table[(1+n)Floor[1+n/2](n!/Floor[1+n/2]!)^2,{n,0,30}] (* _Harvey P. Dale_, Oct 01 2023 *)

%o (Python)

%o def A329965():

%o x, n = 1, 1

%o while true:

%o yield x

%o m = n if n % 2 else 4/(n+2)

%o n += 1

%o x *= m * n

%o a = A329965(); [next(a) for i in range(36)]

%Y Cf. A212303, A057977, A052510, A123072, A093005, A262033, A329964.

%K nonn

%O 0,2

%A _Peter Luschny_, Dec 04 2019