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A203516 a(n) = Product_{1 <= i < j <= n} 2*(i+j-1).

%I A203516 #25 Feb 19 2024 14:54:00
%S A203516 1,4,192,184320,4954521600,4794391461888000,204135216112950312960000,
%T A203516 451965950843675288237663846400000,
%U A203516 60040562704967329457107799785403842560000000,542366306792798635131534558788357929673196306432000000000
%N A203516 a(n) = Product_{1 <= i < j <= n} 2*(i+j-1).
%C A203516 Each term divides its successor, as in A034910.
%C A203516 See A093883 for a guide to related sequences.
%H A203516 G. C. Greubel, <a href="/A203516/b203516.txt">Table of n, a(n) for n = 1..33</a>
%F A203516 a(n) ~ sqrt(A) * 2^(-7/24 - n + 3*n^2/2) * exp(-1/24 + n/2 - 3*n^2/4) * n^(1/24 - n/2 + n^2/2), where A = A074962 is the Glaisher-Kinkelin constant. - _Vaclav Kotesovec_, Sep 01 2023
%F A203516 From _G. C. Greubel_, Feb 19 2024: (Start)
%F A203516 a(n) = BarnesG(n+1)*A203517(n).
%F A203516 a(n) = 2^binomial(n,2) * Product_{j=1..n-1} (2j)!/j!. (End)
%p A203516 a:= n-> mul(mul(2*(i+j-1), i=1..j-1), j=2..n):
%p A203516 seq(a(n), n=1..12);  # _Alois P. Heinz_, Jul 23 2017
%t A203516 f[j_] := 2 j - 1; z = 15;
%t A203516 v[n_] := Product[Product[f[k] + f[j], {j, 1, k - 1}], {k, 2, n}]
%t A203516 d[n_] := Product[(i - 1)!, {i, 1, n}]    (* A000178 *)
%t A203516 Table[v[n], {n, 1, z}]                   (* A203516 *)
%t A203516 Table[v[n + 1]/(4 v[n]), {n, 1, z - 1}]  (* A034910 *)
%t A203516 Table[v[n]/d[n], {n, 1, 20}]             (* A203517 *)
%t A203516 Table[2^(-1/24 - 3*n/2 + 3*n^2/2) * Glaisher^(3/2) * Pi^(1/4 - n/2) * BarnesG[1/2 + n]/E^(1/8), {n, 1, 10}] (* _Vaclav Kotesovec_, Sep 01 2023 *)
%o A203516 (PARI) a(n) = my(pd=1); for(j=1, n, for(i=1, j-1, pd=pd*2*(i+j-1))); pd \\ _Felix Fröhlich_, Jul 23 2017
%o A203516 (Magma) [2^Binomial(n,2)*(&*[Factorial(2*k)/Factorial(k): k in [0..n-1]]): n in [1..20]]; // _G. C. Greubel_, Feb 19 2024
%o A203516 (SageMath) [2^binomial(n,2)*product(factorial(2*k)/factorial(k) for k in range(n)) for n in range(1,21)] # _G. C. Greubel_, Feb 19 2024
%Y A203516 Cf. A000178, A005408, A034910, A093883, A203516, A203517.
%K A203516 nonn
%O A203516 1,2
%A A203516 _Clark Kimberling_, Jan 03 2012
%E A203516 Name edited by _Alois P. Heinz_, Jul 23 2017