A203469 a(n) = v(n)/A000178(n), v = A093883 and A000178 = (superfactorials).

1, 3, 30, 1050, 132300, 61122600, 104886381600, 674943865596000, 16407885372638760000, 1515727634953623371280000, 534621388490302221024396480000, 722849817707190846398223943885440000, 3759035907022704558524683975387453632000000

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, A006963, A076756, A093883, A296590.

Programs

Magma

[(&*[Binomial(2*n-k,k): k in [1..n]]): n in [1..20]]; // G. C. Greubel, Aug 29 2023

Mathematica

(* First program *)
f[j_]:= j; z = 16;
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}]           (* A093883 *)
Table[v[n+1]/v[n], {n,z-1}]  (* A006963 *)
Table[v[n]/d[n], {n,20}]     (* A203469 *)
(* Second program *)
Table[Product[Binomial[2*n-j,j], {j,n}], {n,20}] (* G. C. Greubel, Aug 29 2023 *)

SageMath

[product(binomial(2*n-j,j) for j in range(n)) for n in range(1, 20)] # G. C. Greubel, Aug 29 2023

Formula

a(n) = Product_{i=1..n} binomial(2n-i,i). - Enrique Pérez Herrero, Feb 20 2013

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

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

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

a(n) ~ A^(3/2) * exp(n/2 - 1/8) * 2^(n^2 - 7/24) / (Pi^(n/2 + 1/2) * n^(n/2 + 3/8)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 26 2023