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A203473 a(n) = v(n+1)/v(n), where v=A203472.

%I A203473 #23 Aug 27 2023 01:14:29
%S A203473 7,72,990,17160,360360,8910720,253955520,8204716800,296541907200,
%T A203473 11861676288000,520431047136000,24858235898496000,1284342188088960000,
%U A203473 71382386874839040000,4247252019052922880000
%N A203473 a(n) = v(n+1)/v(n), where v=A203472.
%H A203473 G. C. Greubel, <a href="/A203473/b203473.txt">Table of n, a(n) for n = 1..250</a>
%F A203473 a(n) ~ 2^(2*n + 11/2) * n^n / exp(n). - _Vaclav Kotesovec_, Apr 09 2021
%F A203473 a(n) = RisingFactorial(6 + n, n). - _Peter Luschny_, Mar 22 2022
%F A203473 Since v(n) = (135/4)*(2^(n+2)^2/Pi^(n/2))*(BarnesG(n+3)*BarnesG(n+7/2) )/( BarnesG(9/2)*BarnesG(n+6) ) then v(n+1)/v(n) = Gamma(2*n+6) / Gamma(n+6). - _G. C. Greubel_, Aug 27 2023
%t A203473 (* First program *)
%t A203473 f[j_]:= j+2; z=16;
%t A203473 v[n_]:= Product[Product[f[k] + f[j], {j,k-1}], {k,2,n}];
%t A203473 d[n_]:= Product[(i-1)!, {i,n}]  (* A000178 *)
%t A203473 Table[v[n], {n,z}]              (* A203472 *)
%t A203473 Table[v[n+1]/v[n], {n,z-1}]     (* this sequence *)
%t A203473 Table[v[n]/d[n], {n,20}]        (* A203474 *)
%t A203473 (* Second program *)
%t A203473 Table[Pochhammer[n+6,n], {n,20}] (* _G. C. Greubel_, Aug 27 2023 *)
%o A203473 (Magma) [Floor(Gamma(2*n+6)/Gamma(n+6)): n in [1..16]]; // _G. C. Greubel_, Aug 27 2023
%o A203473 (SageMath) [rising_factorial(n+6, n) for n in range(1,16)] # _G. C. Greubel_, Aug 27 2023
%Y A203473 Cf. A093883, A203472, A203474.
%K A203473 nonn
%O A203473 1,1
%A A203473 _Clark Kimberling_, Jan 02 2012