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A339411 Product of partial sums of odd squares.

%I A339411 #26 Feb 24 2025 08:26:56
%S A339411 1,1,10,350,29400,4851000,1387386000,631260630000,429257228400000,
%T A339411 415950254319600000,553213838245068000000,979741707532015428000000,
%U A339411 2253405927323635484400000000,6591212337421633791870000000000,24084289880938649875492980000000000,108258883014819231190340945100000000000
%N A339411 Product of partial sums of odd squares.
%C A339411 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
%H A339411 Emily H. Dickey and Noah A. Rosenberg, <a href="https://doi.org/10.1098/rstb.2023.0307">Labelled histories with multifurcation and simultaneity</a>, Phil. Trans. R. Soc. B 380 (2025), 20230307.
%F A339411 a(n) = Product_{i=1..n} (Sum_{j=1..i} (2*j - 1)^2).
%F A339411 a(n) = Product_{i=1..n} binomial(2*i + 1, 3).
%F A339411 a(n) = Product_{i=1..n} A000447(i).
%F A339411 a(n) = ((2*n)! * (2*n+1)!) / (n! * 12^n).
%F A339411 a(n) / A135438(n) = A000108(n).
%F A339411 a(n) = (Gamma(2*n + 2)*Gamma(n + 1/2))/(3^n*sqrt(Pi)). - _Peter Luschny_, Dec 11 2020
%F A339411 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 A339411 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 A339411 a:= proc(n) option remember;
%p A339411       `if`(n=0, 1, a(n-1)*(4*n^3-n)/3)
%p A339411     end:
%p A339411 seq(a(n), n=0..15);  # _Alois P. Heinz_, Dec 03 2020
%t A339411 Array[((2 #)!*(2 # + 1)!)/(#!*12^#) &, 16, 0] (* _Michael De Vlieger_, Dec 10 2020 *)
%o A339411 (PARI) for(n=0,9,print((2*n)!*(2*n+1)!/(n!*12^n)))
%o A339411 (PARI) for(n=0,9,print(prod(i=1,n,sum(j=1,i,(2*j-1)^2))))
%Y A339411 Cf. A016754, A000447, A135438, A000108, A006472.
%K A339411 nonn,easy
%O A339411 0,3
%A A339411 _Werner Schulte_, Dec 03 2020