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A203471 a(n) = v(n)/A000178(n), v = A203470, A000178 = (superfactorials).

%I A203471 #18 Oct 31 2024 01:33:40
%S A203471 1,5,105,8820,2910600,3745942200,18748440711000,364619674947528000,
%T A203471 27558684271884061296000,8100324068034882136733280000,
%U A203471 9267305355220395466643896716480000,41308086890359390753018505224037952000000
%N A203471 a(n) = v(n)/A000178(n), v = A203470, A000178 = (superfactorials).
%H A203471 G. C. Greubel, <a href="/A203471/b203471.txt">Table of n, a(n) for n = 1..55</a>
%F A203471 From _G. C. Greubel_, Aug 29 2023: (Start)
%F A203471 a(n) = Product_{j=1..n} Gamma(2*j+2)/(Gamma(j)*Gamma(j+3)).
%F A203471 a(n) = (2/sqrt(Pi))*( 2^(n+1)^2 * BarnesG(n+5/2) /(Pi^(n/2) * Gamma(n+2)*Gamma(n+3)*BarnesG(3/2)*BarnesG(n+1)) ).
%F A203471 a(n) = (BarnesG(n+2)/(2^n * BarnesG(n+1))) * Product_{j=1..n} Catalan(j+1). (End)
%F A203471 a(n) ~ A^(3/2) * 2^(n^2 + 2*n + 41/24) * exp(n/2 - 1/8) / (n^(n/2 + 23/8) * Pi^(n/2 + 1)), where A is the Glaisher-Kinkelin constant A074962. - _Vaclav Kotesovec_, Nov 19 2023
%F A203471 a(n) = Product_{1 <= j <= i <= n-1}  (i + j + 3)/(i - j + 1). - _Peter Bala_, Oct 25 2024
%t A203471 (* First program *)
%t A203471 f[j_]:= j+1; z = 16;
%t A203471 v[n_]:= Product[Product[f[k] + f[j], {j,k-1}], {k,2,n}]
%t A203471 d[n_]:= Product[(i-1)!, {i,n}]
%t A203471 Table[v[n], {n, z}]           (* A203470 *)
%t A203471 Table[v[n+1]/v[n], {n, z-1}]  (* A102693 *)
%t A203471 Table[v[n]/d[n], {n, 20}]     (* A203471 *)
%t A203471 (* Second program *)
%t A203471 Table[Product[Gamma[2*j+2]/(Gamma[j]*Gamma[j+3]), {j,n}], {n,20}] (* _G. C. Greubel_, Aug 29 2023 *)
%o A203471 (Magma) [(&*[Factorial(2*k+1)/(Factorial(k-1)*Factorial(k+2)): k in [1..n]]): n in [1..20]]; // G. C. Greubel, Aug 29 2023
%o A203471 (SageMath) [product(gamma(2*k+4)/(gamma(k+1)*gamma(k+4)) for k in range(n)) for n in range(1, 20)] # G. C. Greubel, Aug 29 2023
%Y A203471 Cf. A000178, A007685, A102693, A203470, A296589, A338550.
%K A203471 nonn
%O A203471 1,2
%A A203471 _Clark Kimberling_, Jan 02 2012