A203470 - OEIS

%I #21 Aug 29 2023 04:24:05

%S 1,5,210,105840,838252800,129459762432000,466521199899955200000,

%T 45727437650097816797184000000,

%U 139352822480378029387123167068160000000,14863555768518278744824500982673408262144000000000,61707340455179609358720715109663452970925870494515200000000000

%N a(n) = Product_{2 <= i < j <= n+1} (i + j).

%C Each term divides its successor, as (conjectured) in A102693.  Each term is divisible by the corresponding superfactorial, A000178(n), as in A203471.

%H G. C. Greubel, <a href="/A203470/b203470.txt">Table of n, a(n) for n = 1..36</a>

%F a(n) ~ sqrt(A) * 2^(n^2 + 5*n/2 + 41/24) * exp(-3*n^2/4 + n/2 - 1/24) * n^(n^2/2 - n/2 - 71/24) / Pi, where A is the Glaisher-Kinkelin constant A074962. - _Vaclav Kotesovec_, Apr 08 2021

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

%F a(n) = Product_{j=2..n+1} Gamma(2*j)/Gamma(j+2).

%F a(n) = (2/sqrt(Pi))*( 2^(n+1)^2 * BarnesG(n+5/2)/(Pi^(n/2)*Gamma(n+2)*Gamma(n+3)*BarnesG(3/2)) ).

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

%p a:= n-> mul(mul(i+j, i=2..j-1), j=3..n+1):

%p seq(a(n), n=1..12);  # _Alois P. Heinz_, Jul 23 2017

%t (* First program *)

%t f[j_]:= j+1; 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}]           (* A203470 *)

%t Table[v[n+1]/v[n], {n, z-1}]  (* A102693 *)

%t Table[v[n]/d[n], {n, 20}]     (* A203471 *)

%t (* Second program *)

%t Table[Product[Gamma[2*j]/Gamma[j+2], {j,2,n+1}], {n,20}] (* _G. C. Greubel_, Aug 29 2023 *)

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

%o (SageMath) [product(gamma(2*k)/gamma(k+2) for k in range(2,n+2)) for n in range(1,20)] # _G. C. Greubel_, Aug 29 2023

%Y Cf. A000178, A102693, A203471.

%K nonn

%O 1,2

%A _Clark Kimberling_, Jan 02 2012

%E Name edited by _Alois P. Heinz_, Jul 23 2017