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A203434 a(n) = A203433(n)/A000178(n) where A000178=(superfactorials).

%I A203434 #17 May 27 2025 11:11:09
%S A203434 1,1,3,6,45,189,3402,30618,1299078,25332021,2507870079,106698472452,
%T A203434 24487299427734,2283997201168644,1209640056157393380,
%U A203434 248218139523497121576,302358334494179897593596,136861610819571430116630660
%N A203434 a(n) = A203433(n)/A000178(n) where A000178=(superfactorials).
%H A203434 G. C. Greubel, <a href="/A203434/b203434.txt">Table of n, a(n) for n = 1..100</a>
%H A203434 R. Chapman, <a href="https://web.archive.org/web/20230227162446/https://www.maa.org/sites/default/files/Robin_Chapman27238.pdf">A polynomial taking integer values</a>, Mathematics Magazine, 29 (1996), 121.
%t A203434 f[j_]:= j + Floor[(j+1)/2]; z = 20;
%t A203434 v[n_]:= Product[Product[f[k] - f[j], {j,k-1}], {k,2,n}]
%t A203434 d[n_]:= Product[(i-1)!, {i,n}]
%t A203434 Table[v[n], {n,z}]             (* A203433 *)
%t A203434 Table[v[n+1]/v[n], {n,z}]      (* A014402 *)
%t A203434 Table[v[n]/d[n], {n,z}]        (* A203434 *)
%o A203434 (Magma)
%o A203434 Barnes:= func< n | (&*[Factorial(j): j in [1..n-1]]) >;
%o A203434 f:= func< k | (&*[k+1-j+Floor((k+2)/2)-Floor((j+1)/2): j in [1..k]]) >;
%o A203434 [1] cat [(&*[f(k): k in [1..n-1]])/Barnes(n): n in [2..20]]; // _G. C. Greubel_, Sep 19 2023
%o A203434 (SageMath)
%o A203434 def barnes(n): return product(factorial(j) for j in range(n))
%o A203434 def f(k): return product(k-j+(k//2)-(j//2) for j in range(k))
%o A203434 [product(f(k) for k in range(1, n) )//barnes(n) for n in range(1,31)] # _G. C. Greubel_, Sep 19 2023
%Y A203434 Cf. A000178, A007494, A014402, A203432, A203433.
%K A203434 nonn
%O A203434 1,3
%A A203434 _Clark Kimberling_, Jan 02 2012