A203474 - OEIS

%I #20 Aug 27 2023 04:33:05

%S 1,7,252,41580,29729700,89278289100,1104908105901600,

%T 55674109640169820800,11329124570678156834592000,

%U 9258047307912482983660236480000,30262334718212007877669234596364800000

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

%H G. C. Greubel, <a href="/A203474/b203474.txt">Table of n, a(n) for n = 1..55</a>

%F 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

%F From _G. C. Greubel_, Aug 27 2023: (Start)

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

%F 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)

%t (* First program *)

%t f[j_]:= j+2; 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}]  (* A000178(n-1) *)

%t Table[v[n], {n,z}]              (* A203472 *)

%t Table[v[n+1]/v[n], {n,z-1}]     (* A203473 *)

%t Table[v[n]/d[n], {n,20}]        (* A203474 *)

%t (* Second program *)

%t Table[Product[Binomial[2*j+3, j+4], {j,n}], {n,20}] (* _G. C. Greubel_, Aug 27 2023 *)

%o (Magma) [(&*[ Binomial(2*j+3, j+4): j in [1..n]]): n in [1..20]]; // _G. C. Greubel_, Aug 27 2023

%o (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

%Y Cf. A000178, A093883, A203472, A203473.

%K nonn

%O 1,2

%A _Clark Kimberling_, Jan 02 2012

%E Definition corrected by _Vaclav Kotesovec_, Apr 09 2021