cp's OEIS Frontend

A203470 a(n) = Product_{2 <= i < j <= n+1} (i + j).

%I A203470 #25 Aug 27 2025 03:57:49
%S A203470 1,5,210,105840,838252800,129459762432000,466521199899955200000,
%T A203470 45727437650097816797184000000,
%U A203470 139352822480378029387123167068160000000,14863555768518278744824500982673408262144000000000,61707340455179609358720715109663452970925870494515200000000000
%N A203470 a(n) = Product_{2 <= i < j <= n+1} (i + j).
%C A203470 Each term divides its successor, as (conjectured) in A102693.  Each term is divisible by the corresponding superfactorial, A000178(n), as in A203471.
%H A203470 G. C. Greubel, <a href="/A203470/b203470.txt">Table of n, a(n) for n = 1..36</a>
%F A203470 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 A203470 From _G. C. Greubel_, Aug 29 2023: (Start)
%F A203470 a(n) = Product_{j=2..n+1} Gamma(2*j)/Gamma(j+2).
%F A203470 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 A203470 a(n) = (BarnesG(n+2)/2^n) * Product_{j=2..n+1} Catalan(j). (End)
%p A203470 a:= n-> mul(mul(i+j, i=2..j-1), j=3..n+1):
%p A203470 seq(a(n), n=1..12);  # _Alois P. Heinz_, Jul 23 2017
%t A203470 (* First program *)
%t A203470 f[j_]:= j+1; z = 16;
%t A203470 v[n_]:= Product[Product[f[k] + f[j], {j,k-1}], {k,2,n}]
%t A203470 d[n_]:= Product[(i-1)!, {i,n}]
%t A203470 Table[v[n], {n, z}]           (* A203470 *)
%t A203470 Table[v[n+1]/v[n], {n, z-1}]  (* A102693 *)
%t A203470 Table[v[n]/d[n], {n, 20}]     (* A203471 *)
%t A203470 (* Second program *)
%t A203470 Table[Product[Gamma[2*j]/Gamma[j+2], {j,2,n+1}], {n,20}] (* _G. C. Greubel_, Aug 29 2023 *)
%o A203470 (Magma) [(&*[Factorial(2*k-1)/Factorial(k+1): k in [2..n+1]]): n in [1..20]]; // _G. C. Greubel_, Aug 29 2023
%o A203470 (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
%o A203470 (Python)
%o A203470 from math import prod, factorial
%o A203470 def A203470(n): return prod(factorial(2*k+1)//factorial(k+2) for k in range(1,n+1)) # _Chai Wah Wu_, Aug 26 2025
%Y A203470 Cf. A000178, A102693, A203471.
%K A203470 nonn,changed
%O A203470 1,2
%A A203470 _Clark Kimberling_, Jan 02 2012
%E A203470 Name edited by _Alois P. Heinz_, Jul 23 2017