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%I #24 Sep 01 2023 07:16:18
%S 6,3024,2874009600,159950125679984640000,
%T 20708778572935434707683938140160000000,
%U 302101709923756073800654275737927385319576932502732800000000000
%N Product of all possible sums of three numbers taken from among first n natural numbers.
%D Amarnath Murthy, Another combinatorial approach towards generalizing the AM GM inequality, Octogon Mathematical Magazine Vol. 8, No. 2, October 2000.
%D Amarnath Murthy, Smarandache Dual Symmetric Functions And Corresponding Numbers Of The Type Of Stirling Numbers Of The First Kind. Smarandache Notions Journal Vol. 11, No. 1-2-3 Spring 2000.
%H Vaclav Kotesovec, <a href="/A093884/b093884.txt">Table of n, a(n) for n = 3..17</a>
%F a(n) ~ sqrt(Pi/A) * 2^(5/12 - n/4 - n^2 - 2*n^3/3) * 3^(-1/6 - 7*n/24 + 3*n^3/4) * exp(1/24 - n/3 + 3*n^2/4 - 11*n^3/36 + zeta(3)/(48*Pi^2)) * n^(11/24 + n/3 - n^2/2 + n^3/6), where A = A074962 is the Glaisher-Kinkelin constant. - _Vaclav Kotesovec_, Aug 31 2023
%e a(4) = (1+2+3)*(1+2+4)*(1+3+4)*(2+3+4) = 3024.
%t Table[Product[(j + k + m), {k, 2, n}, {j, 1, k - 1}, {m, 1, j - 1}], {n, 3, 10}] (* _Vaclav Kotesovec_, Aug 31 2023 *)
%t Table[Product[Sqrt[BarnesG[3*k] * BarnesG[k+2] * Gamma[k/2 + 1] / Gamma[3*k/2]] / (BarnesG[2*k + 1] * 2^((k-1)/2)), {k, 1, n}], {n, 3, 10}] (* _Vaclav Kotesovec_, Aug 31 2023 *)
%Y Cf. A093883, A306594.
%K nonn
%O 3,1
%A _Amarnath Murthy_, Apr 22 2004
%E More terms from _Vladeta Jovovic_, May 27 2004