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A203467 a(n) = A203309(n)/A000178(n) where A000178 are superfactorials.

%I A203467 #16 Nov 19 2023 06:10:52
%S A203467 1,1,2,15,630,198450,589396500,19912024006875,8969371213896843750,
%T A203467 61815874928487448987968750,7358663747680777931818630148437500,
%U A203467 16862758880642741957030086746987589746093750
%N A203467 a(n) = A203309(n)/A000178(n) where A000178 are superfactorials.
%H A203467 G. C. Greubel, <a href="/A203467/b203467.txt">Table of n, a(n) for n = 0..39</a>
%H A203467 R. Chapman, <a href="https://www.jstor.org/stable/2690667">A polynomial taking integer values</a>, Mathematics Magazine, 29 (1996), 121.
%F A203467 From _G. C. Greubel_, Aug 29 2023: (Start)
%F A203467 a(n) = (2^(n+3)/Pi)^(n/2)*BarnesG(n+3/2)/(Gamma(n+ 2)*BarnesG(3/2)).
%F A203467 a(n) = (1/2)^binomial(n,2)*BarnesG(n+1)*Product_{k=2..n} binomial(2*k, k+1).
%F A203467 a(n) = Product_{k=1..n-1} (2*k+2)!/(2^k*(k+2)!). (End)
%F A203467 a(n) ~ sqrt(A/Pi) * 2^(n^2/2 + 2*n - 7/24) * n^(n^2/2 - n/2 - 35/24) / exp(3*n^2/4 - n/2 + 1/24), where A is the Glaisher-Kinkelin constant A074962. - _Vaclav Kotesovec_, Nov 19 2023
%t A203467 (* First program *)
%t A203467 f[j_]:= j*(j+1)/2; z = 15;
%t A203467 v[n_]:= Product[Product[f[k] - f[j], {j,k-1}], {k,2,n}]
%t A203467 d[n_]:= Product[(i-1)!, {i,n}]
%t A203467 Table[v[n], {n,0,z}]           (* A203309 *)
%t A203467 Table[v[n+1]/v[n], {n,z}]      (* A203310 *)
%t A203467 Table[v[n]/d[n], {n,0,12}]     (* A203467 *)
%t A203467 (* Second program *)
%t A203467 Table[Product[(2*k+2)!/(2^k*(k+2)!), {k,n-1}], {n,0,20}] (* _G. C. Greubel_, Aug 29 2023 *)
%o A203467 (Magma) F:= Factorial; [1] cat [(&*[(F(2*k+2))/(2^k*F(k+2)): k in [0..n]]): n in [0..20]]; // _G. C. Greubel_, Aug 29 2023
%o A203467 (SageMath) f=factorial; [product((f(2*j+2))/(2^j*f(j+2)) for j in range(n)) for n in range(21)] # _G. C. Greubel_, Aug 29 2023
%Y A203467 Cf. A000178, A203309, A203310.
%K A203467 nonn
%O A203467 0,3
%A A203467 _Clark Kimberling_, Jan 02 2012
%E A203467 Name edited by _Michel Marcus_, May 17 2019
%E A203467 a(0) = 1 prepended by _G. C. Greubel_, Aug 29 2023