A339411 Product of partial sums of odd squares.

1, 1, 10, 350, 29400, 4851000, 1387386000, 631260630000, 429257228400000, 415950254319600000, 553213838245068000000, 979741707532015428000000, 2253405927323635484400000000, 6591212337421633791870000000000, 24084289880938649875492980000000000, 108258883014819231190340945100000000000

Offset

0,3

Links

Table of n, a(n) for n=0..15.

Formula

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

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

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

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

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

a(n) = (Gamma(2*n + 2)*Gamma(n + 1/2))/(3^n*sqrt(Pi)). - Peter Luschny, Dec 11 2020

D-finite with recurrence 3*a(n) -n*(2*n-1)*(2*n+1)*a(n-1)=0. - R. J. Mathar, Jan 25 2023

Example

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.

Maple

a:= proc(n) option remember;
      `if`(n=0, 1, a(n-1)*(4*n^3-n)/3)
    end:
seq(a(n), n=0..15);  # Alois P. Heinz, Dec 03 2020

Mathematica

Array[((2 #)!*(2 # + 1)!)/(#!*12^#) &, 16, 0] (* Michael De Vlieger, Dec 10 2020 *)

Prog

(PARI) for(n=0, 9, print((2*n)!*(2*n+1)!/(n!*12^n)))
(PARI) for(n=0, 9, print(prod(i=1, n, sum(j=1, i, (2*j-1)^2))))

Crossrefs

Cf. A016754, A000447, A135438, A000108.

Sequence in context: A060704 A220185 A132093 * A217507 A217508 A217509

Adjacent sequences: A339408 A339409 A339410 * A339412 A339413 A339414

Keyword

nonn,easy

Author

Werner Schulte, Dec 03 2020

Status

approved