A203432 - OEIS

%I #18 Sep 28 2023 01:59:36

%S 1,2,3,15,45,540,3402,96228,1299078,85739148,2507870079,383704122087,

%T 24487299427734,8645900336407620,1209640056157393380,

%U 982320774834892454820,302358334494179897593596,563293577162657149216869348

%N a(n) = A203430(n)/A000178(n) where A000178=(superfactorials).

%H G. C. Greubel, <a href="/A203432/b203432.txt">Table of n, a(n) for n = 1..100</a>

%H R. Chapman, <a href="https://www.maa.org/sites/default/files/Robin_Chapman27238.pdf">A polynomial taking integer values</a>, Mathematics Magazine, 29 (1996), 121.

%t f[j_]:= j + Floor[j/2]; z = 20;

%t v[n_]:= Product[Product[f[k] - f[j], {j,k-1}], {k,2,n}]

%t d[n_]:= Product[(i-1)!, {i,n}]

%t Table[v[n], {n,z}]             (* A203430 *)

%t Table[v[n+1]/v[n], {n,z}]      (* A203431 *)

%t Table[v[n]/d[n], {n,z}]        (* this sequence *)

%o (Magma)

%o Barnes:= func< n | (&*[Factorial(j): j in [1..n-1]]) >;

%o 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(n): n in [1..25]]; // _G. C. Greubel_, Sep 20 2023

%o (SageMath)

%o def barnes(n): return product(factorial(j) for j in range(n))

%o 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(n) for n in range(1,31)] # _G. C. Greubel_, Sep 20 2023

%Y Cf. A000178, A032766, A203430, A203431.

%K nonn

%O 1,2

%A _Clark Kimberling_, Jan 02 2012