A165970 - OEIS

%I #20 Jul 05 2019 16:43:34

%S 1,12,14515200,420505587390873600000,

%T 6848282921689337839624757371207680000000000,

%U 592617982969061328644755583860005865281724398591341934673920000000000000000

%N a(n) = sqrt( superfactorial(4n) / factorial(2n) ).

%C For n>=5, 2^(12*n)*10^(12*(n - 4)) | a(n). - _G. C. Greubel_, Apr 18 2016

%H Seiichi Manyama, <a href="/A165970/b165970.txt">Table of n, a(n) for n = 0..14</a>

%F a(n) = sqrt( A000178(4n) / A000142(2n) ) = sqrt(0! * 1! * ... * (2n-1)! * (2n+1)! * (2n+2)! * ... * (4n)!).

%F a(n) ~ 2^(8*n^2 + 4*n + 1/6) * n^(4*n^2 + n - 1/24) * Pi^n / (A^(1/2) * exp(6*n^2 + n - 1/24)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - _Vaclav Kotesovec_, Jul 10 2015

%F a(n) = 2^n * Product_{k=1..2*n} (2*k-1)!. - _Seiichi Manyama_, Jul 05 2019

%F a(n) = A^(3/2) * exp(-1/8) * 2^(4*n^2 + n - 1/24) * BarnesG(2*n + 3/2) * BarnesG(2*n + 1) / Pi^(n + 1/4), where A is the Glaisher-Kinkelin constant A074962. - _Vaclav Kotesovec_, Jul 05 2019

%t Table[Sqrt[Product[k!,{k,0,4*n}]/(2*n)!],{n,0,10}] (* _Vaclav Kotesovec_, Jul 10 2015 *)

%o (PARI) {a(n) = 2^n*prod(k=1, 2*n, (2*k-1)!)} \\ _Seiichi Manyama_, Jul 05 2019

%Y Cf. A168467.

%K nonn

%O 0,2

%A _Max Alekseyev_, Oct 02 2009