A203474 a(n) = A203472(n) / A000178(n-1), where A000178 are the superfactorials.

1, 7, 252, 41580, 29729700, 89278289100, 1104908105901600, 55674109640169820800, 11329124570678156834592000, 9258047307912482983660236480000, 30262334718212007877669234596364800000

Offset: 1

Author

Clark Kimberling, Jan 02 2012

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 1..55

Crossrefs

Cf. A000178, A093883, A203472, A203473.

Programs

Magma

[(&*[ Binomial(2*j+3, j+4): j in [1..n]]): n in [1..20]]; // G. C. Greubel, Aug 27 2023

Mathematica

(* First program *)
f[j_]:= j+2; z=16;
v[n_]:= Product[Product[f[k] + f[j], {j,k-1}], {k,2,n}];
d[n_]:= Product[(i-1)!, {i,n}]  (* A000178(n-1) *)
Table[v[n], {n,z}]              (* A203472 *)
Table[v[n+1]/v[n], {n,z-1}]     (* A203473 *)
Table[v[n]/d[n], {n,20}]        (* A203474 *)
(* Second program *)
Table[Product[Binomial[2*j+3, j+4], {j,n}], {n,20}] (* G. C. Greubel, Aug 27 2023 *)

SageMath

[product( binomial(2*j+5,j+5) for j in range(n) ) for n in range(1,20)] # G. C. Greubel, Aug 27 2023

Formula

a(n) ~ 3*A^(3/2) * 2^(n^2 + 4*n + 185/24) * exp(n/2 - 1/8) / (Pi^(n/2 + 3/2) * n^(n/2 + 59/8)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 09 2021

From G. C. Greubel, Aug 27 2023: (Start)

a(n) = Product_{j=1..n} binomial(2*j+3, j+4).

a(n) = (18*2^(n+2)^2/Pi^(n/2))*BarnesG(n+3)*BarnesG(n+7/2)/( BarnesG(n +1)*BarnesG(n+6)*BarnesG(7/2)). (End)

Extensions

Definition corrected by Vaclav Kotesovec, Apr 09 2021