A339411 - OEIS

%I #18 Jan 25 2023 10:43:25

%S 1,1,10,350,29400,4851000,1387386000,631260630000,429257228400000,

%T 415950254319600000,553213838245068000000,979741707532015428000000,

%U 2253405927323635484400000000,6591212337421633791870000000000,24084289880938649875492980000000000,108258883014819231190340945100000000000

%N Product of partial sums of odd squares.

%F a(n) = Product_{i=1..n} (Sum_{j=1..i} (2*j - 1)^2).

%F a(n) = Product_{i=1..n} binomial(2*i + 1, 3).

%F a(n) = Product_{i=1..n} A000447(i).

%F a(n) = ((2*n)! * (2*n+1)!) / (n! * 12^n).

%F a(n) / A135438(n) = A000108(n).

%F a(n) = (Gamma(2*n + 2)*Gamma(n + 1/2))/(3^n*sqrt(Pi)). - _Peter Luschny_, Dec 11 2020

%F D-finite with recurrence 3*a(n) -n*(2*n-1)*(2*n+1)*a(n-1)=0. - _R. J. Mathar_, Jan 25 2023

%e a(4) = (1^2) * (1^2 + 3^2) * (1^2 + 3^2 + 5^2) * (1^2 + 3^2 + 5^2 + 7^2) = 1 * 10 * 35 * 84 = 29400.

%p a:= proc(n) option remember;

%p       `if`(n=0, 1, a(n-1)*(4*n^3-n)/3)

%p     end:

%p seq(a(n), n=0..15);  # _Alois P. Heinz_, Dec 03 2020

%t Array[((2 #)!*(2 # + 1)!)/(#!*12^#) &, 16, 0] (* _Michael De Vlieger_, Dec 10 2020 *)

%o (PARI) for(n=0,9,print((2*n)!*(2*n+1)!/(n!*12^n)))

%o (PARI) for(n=0,9,print(prod(i=1,n,sum(j=1,i,(2*j-1)^2))))

%Y Cf. A016754, A000447, A135438, A000108.

%K nonn,easy

%O 0,3

%A _Werner Schulte_, Dec 03 2020