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A203432 a(n) = A203430(n)/A000178(n) where A000178=(superfactorials).

%I A203432 #21 May 27 2025 11:11:14
%S A203432 1,2,3,15,45,540,3402,96228,1299078,85739148,2507870079,383704122087,
%T A203432 24487299427734,8645900336407620,1209640056157393380,
%U A203432 982320774834892454820,302358334494179897593596,563293577162657149216869348
%N A203432 a(n) = A203430(n)/A000178(n) where A000178=(superfactorials).
%H A203432 G. C. Greubel, <a href="/A203432/b203432.txt">Table of n, a(n) for n = 1..100</a>
%H A203432 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 A203432 f[j_]:= j + Floor[j/2]; z = 20;
%t A203432 v[n_]:= Product[Product[f[k] - f[j], {j,k-1}], {k,2,n}]
%t A203432 d[n_]:= Product[(i-1)!, {i,n}]
%t A203432 Table[v[n], {n,z}]             (* A203430 *)
%t A203432 Table[v[n+1]/v[n], {n,z}]      (* A203431 *)
%t A203432 Table[v[n]/d[n], {n,z}]        (* this sequence *)
%o A203432 (Magma)
%o A203432 Barnes:= func< n | (&*[Factorial(j): j in [1..n-1]]) >;
%o A203432 A203432:= func< n | n eq 1 select 1 else (&*[(&*[k-j+Floor((k+1)/2)-Floor((j+1)/2): j in [0..k-1]]) : k in [1..n-1]])/Barnes(n) >;
%o A203432 [A203432(n): n in [1..25]]; // _G. C. Greubel_, Sep 20 2023
%o A203432 (SageMath)
%o A203432 def barnes(n): return product(factorial(j) for j in range(n))
%o A203432 def A203432(n): return product(product(k-j+((k+1)//2)-((j+1)//2) for j in range(k)) for k in range(1,n))/barnes(n)
%o A203432 [A203432(n) for n in range(1,31)] # _G. C. Greubel_, Sep 20 2023
%Y A203432 Cf. A000178, A032766, A203430, A203431.
%K A203432 nonn
%O A203432 1,2
%A A203432 _Clark Kimberling_, Jan 02 2012