A203469 - OEIS

%I #29 Nov 26 2023 04:19:43

%S 1,3,30,1050,132300,61122600,104886381600,674943865596000,

%T 16407885372638760000,1515727634953623371280000,

%U 534621388490302221024396480000,722849817707190846398223943885440000,3759035907022704558524683975387453632000000

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

%H G. C. Greubel, <a href="/A203469/b203469.txt">Table of n, a(n) for n = 1..55</a>

%F a(n) = Product_{i=1..n} binomial(2n-i,i).  - _Enrique Pérez Herrero_, Feb 20 2013

%F From _G. C. Greubel_, Aug 29 2023: (Start)

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

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

%F 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

%t (* First program *)

%t f[j_]:= j; z = 16;

%t v[n_]:= Product[Product[f[k] + f[j], {j,k-1}], {k,2,n}]

%t d[n_]:= Product[(i-1)!, {i,n}]

%t Table[v[n], {n,z}]           (* A093883 *)

%t Table[v[n+1]/v[n], {n,z-1}]  (* A006963 *)

%t Table[v[n]/d[n], {n,20}]     (* A203469 *)

%t (* Second program *)

%t Table[Product[Binomial[2*n-j,j], {j,n}], {n,20}] (* _G. C. Greubel_, Aug 29 2023 *)

%o (Magma) [(&*[Binomial(2*n-k,k): k in [1..n]]): n in [1..20]]; // _G. C. Greubel_, Aug 29 2023

%o (SageMath) [product(binomial(2*n-j,j) for j in range(n)) for n in range(1, 20)] # _G. C. Greubel_, Aug 29 2023

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

%K nonn

%O 1,2

%A _Clark Kimberling_, Jan 02 2012