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A203472 a(n) = Product_{3 <= i < j <= n+2} (i + j).

%I A203472 #19 Aug 26 2023 11:53:57
%S A203472 1,7,504,498960,8562153600,3085457671296000,27493649380770693120000,
%T A203472 6982164025191299372050022400000,
%U A203472 57286678477842677171688269225656320000000,16987900892972660430046341200043192304533504000000000,201504981205067832055356568153709798734509139298353152000000000000
%N A203472 a(n) = Product_{3 <= i < j <= n+2} (i + j).
%C A203472 Each term divides its successor, as in A203470. It is conjectured that each term is divisible by the corresponding superfactorial, A000178(n), as in A203474.
%H A203472 G. C. Greubel, <a href="/A203472/b203472.txt">Table of n, a(n) for n = 1..35</a>
%F A203472 a(n) ~ 3*sqrt(A) * 2^(n^2 + 9*n/2 + 185/24) * n^(n^2/2 - n/2 - 179/24) / (Pi^(3/2) * exp(3*n^2/4 - n/2 + 1/24)), where A is the Glaisher-Kinkelin constant A074962. - _Vaclav Kotesovec_, Apr 09 2021
%F A203472 From _G. C. Greubel_, Aug 26 2023: (Start)
%F A203472 a(n) = Prod_{j=3..n+2} Prod_{i=3..j-1} (i + j).
%F A203472 a(n) = Prod_{j=3..n+2} Gamma(2*j)/Gamma(j+3).
%F A203472 a(n) = (18*2^(n+2)^2/Pi^(n/2))*BarnesG(n+3)*BarnesG(n+7/2)/(BarnesG(n+ 6)*BarnesG(7/2)). (End)
%p A203472 a:= n-> mul(mul(i+j, i=3..j-1), j=4..n+2):
%p A203472 seq(a(n), n=1..12);  # _Alois P. Heinz_, Jul 23 2017
%t A203472 (* First program *)
%t A203472 f[j_]:= j + 2;    z = 16;
%t A203472 v[n_]:= Product[Product[f[k] + f[j], {j,k-1}], {k,2,n}]
%t A203472 d[n_]:= Product[(i-1)!, {i,n}] (* A000178 *)
%t A203472 Table[v[n], {n,z}]             (* A203472 *)
%t A203472 Table[v[n+1]/v[n], {n,z-1}]    (* A203473 *)
%t A203472 Table[v[n]/d[n], {n,20}]       (* A203474 *)
%t A203472 (* Second program *)
%t A203472 Table[(18*2^(n+2)^2/Pi^(n/2))*BarnesG[n+3]*BarnesG[n+7/2]/(BarnesG[n+ 6]*BarnesG[7/2]), {n,20}] (* _G. C. Greubel_, Aug 26 2023 *)
%o A203472 (Magma) [(&*[(&*[i+j: i in [3..j]])/(2*j): j in [3..n+2]]): n in [1..20]]; // _G. C. Greubel_, Aug 26 2023
%o A203472 (SageMath) [product( gamma(2*j)/gamma(j+3) for j in range(3,n+3) ) for n in range(1,20)] # _G. C. Greubel_, Aug 26 2023
%Y A203472 Cf. A000178, A093883, A203470, A203473, A203474.
%K A203472 nonn
%O A203472 1,2
%A A203472 _Clark Kimberling_, Jan 02 2012
%E A203472 Name edited by _Alois P. Heinz_, Jul 23 2017