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

Author

Werner Schulte, Dec 03 2020

Keywords

• nonn
• easy

Comments

a(n) is also the number of labeled histories across all trifurcating labeled topologies on trees with non-simultaneous trifurcations, where the number of leaves is 2n+1. - Noah A Rosenberg, Feb 24 2025

Examples

			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.
		

Links

• Emily H. Dickey and Noah A. Rosenberg, Labelled histories with multifurcation and simultaneity, Phil. Trans. R. Soc. B 380 (2025), 20230307.

Crossrefs

Cf. A016754, A000447, A135438, A000108, A006472.

Programs

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 *)

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))))

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