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

1,2

Links

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

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

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 *)

Prog

(Magma) [(&*[Binomial(2*n-k, k): k in [1..n]]): n in [1..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

Crossrefs

Cf. A000178, A006963, A076756, A093883, A296590.

Sequence in context: A184575 A229373 A355400 * A296590 A203478 A012008

Adjacent sequences: A203466 A203467 A203468 * A203470 A203471 A203472

Keyword

nonn

Author

Clark Kimberling, Jan 02 2012

Status

approved