A203467 a(n) = A203309(n)/A000178(n) where A000178 are superfactorials.

1, 1, 2, 15, 630, 198450, 589396500, 19912024006875, 8969371213896843750, 61815874928487448987968750, 7358663747680777931818630148437500, 16862758880642741957030086746987589746093750

Offset: 0

Author

Clark Kimberling, Jan 02 2012

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..39
• R. Chapman, A polynomial taking integer values, Mathematics Magazine, 29 (1996), 121.

Crossrefs

Cf. A000178, A203309, A203310.

Programs

Magma

F:= Factorial; [1] cat [(&*[(F(2*k+2))/(2^k*F(k+2)): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 29 2023

Mathematica

(* First program *)
f[j_]:= j*(j+1)/2; z = 15;
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,0,z}]           (* A203309 *)
Table[v[n+1]/v[n], {n,z}]      (* A203310 *)
Table[v[n]/d[n], {n,0,12}]     (* A203467 *)
(* Second program *)
Table[Product[(2*k+2)!/(2^k*(k+2)!), {k,n-1}], {n,0,20}] (* G. C. Greubel, Aug 29 2023 *)

SageMath

f=factorial; [product((f(2*j+2))/(2^j*f(j+2)) for j in range(n)) for n in range(21)] # G. C. Greubel, Aug 29 2023

Formula

From G. C. Greubel, Aug 29 2023: (Start)

a(n) = (2^(n+3)/Pi)^(n/2)*BarnesG(n+3/2)/(Gamma(n+ 2)*BarnesG(3/2)).

a(n) = (1/2)^binomial(n,2)*BarnesG(n+1)*Product_{k=2..n} binomial(2*k, k+1).

a(n) = Product_{k=1..n-1} (2*k+2)!/(2^k*(k+2)!). (End)

a(n) ~ sqrt(A/Pi) * 2^(n^2/2 + 2*n - 7/24) * n^(n^2/2 - n/2 - 35/24) / exp(3*n^2/4 - n/2 + 1/24), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 19 2023

Extensions

Name edited by Michel Marcus, May 17 2019

a(0) = 1 prepended by G. C. Greubel, Aug 29 2023