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

1, 1, 3, 6, 45, 189, 3402, 30618, 1299078, 25332021, 2507870079, 106698472452, 24487299427734, 2283997201168644, 1209640056157393380, 248218139523497121576, 302358334494179897593596, 136861610819571430116630660

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, A007494, A014402, A203432, A203433.

Programs

Magma

Barnes:= func< n | (&*[Factorial(j): j in [1..n-1]]) >;
f:= func< k | (&*[k+1-j+Floor((k+2)/2)-Floor((j+1)/2): j in [1..k]]) >;
[1] cat [(&*[f(k): k in [1..n-1]])/Barnes(n): n in [2..20]]; // G. C. Greubel, Sep 19 2023

Mathematica

f[j_]:= j + Floor[(j+1)/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}]             (* A203433 *)
Table[v[n+1]/v[n], {n,z}]      (* A014402 *)
Table[v[n]/d[n], {n,z}]        (* A203434 *)

SageMath

def barnes(n): return product(factorial(j) for j in range(n))
def f(k): return product(k-j+(k//2)-(j//2) for j in range(k))
[product(f(k) for k in range(1, n) )//barnes(n) for n in range(1,31)] # G. C. Greubel, Sep 19 2023