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

1, 2, 3, 15, 45, 540, 3402, 96228, 1299078, 85739148, 2507870079, 383704122087, 24487299427734, 8645900336407620, 1209640056157393380, 982320774834892454820, 302358334494179897593596, 563293577162657149216869348

Offset: 1

Author

Clark Kimberling, Jan 02 2012

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 1..100
• R. Chapman, A polynomial taking integer values, Mathematics Magazine, 29 (1996), 121.

Crossrefs

Cf. A000178, A032766, A203430, A203431.

Programs

Magma

Barnes:= func< n | (&*[Factorial(j): j in [1..n-1]]) >;
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) >;
[A203432(n): n in [1..25]]; // G. C. Greubel, Sep 20 2023

Mathematica

f[j_]:= j + Floor[j/2]; z = 20;
v[n_]:= Product[Product[f[k] - f[j], {j,k-1}], {k,2,n}]
d[n_]:= Product[(i-1)!, {i,n}]
Table[v[n], {n,z}]             (* A203430 *)
Table[v[n+1]/v[n], {n,z}]      (* A203431 *)
Table[v[n]/d[n], {n,z}]        (* this sequence *)

SageMath

def barnes(n): return product(factorial(j) for j in range(n))
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)
[A203432(n) for n in range(1,31)] # G. C. Greubel, Sep 20 2023