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A203510 a(n) = A203482(n) / A000178(n).

%I A203510 #24 Feb 24 2024 22:04:34
%S A203510 1,3,84,273000,3046699656000,5996663814749677445376000,
%T A203510 160771799453017261771769947549079938007040000,
%U A203510 6351968589735888467306807912855132014808202373395298410963148996608000000
%N A203510 a(n) = A203482(n) / A000178(n).
%C A203510 It is conjectured that every term of the sequence is an integer.
%H A203510 G. C. Greubel, <a href="/A203510/b203510.txt">Table of n, a(n) for n = 1..16</a>
%F A203510 a(n) ~ c * A * n^(n^3/3 - n^2/4 - 7*n/12 + 17/24) * (2*Pi)^(n^2/4 - 3*n/4) / exp(4*n^3/9 - 7*n^2/8 - n + 1/12), where A is the Glaisher-Kinkelin constant A074962 and c = 0.488888619502150098591650327163991582267254151817880403495924251381414248582... (from A203482). - _Vaclav Kotesovec_, Nov 20 2023
%t A203510 f[j_] := j!; z = 10;
%t A203510 v[n_] := Product[Product[f[k] + f[j], {j, 1, k - 1}], {k, 2, n}]
%t A203510 d[n_] := Product[(i - 1)!, {i, 1, n}]  (* A000178 *)
%t A203510 Table[v[n], {n, 1, z}]                 (* A203482 *)
%t A203510 Table[v[n + 1]/v[n], {n, 1, z - 1}]    (* A203483 *)
%t A203510 Table[v[n]/d[n], {n, 1, 10}]           (* this sequence *)
%t A203510 Table[Product[j! + k!, {j, 1, n}, {k, 1, j-1}] / BarnesG[n+1], {n, 1, 10}] (* _Vaclav Kotesovec_, Nov 20 2023 *)
%o A203510 (Magma)
%o A203510 BarnesG:= func< n | (&*[Factorial(k): k in [0..n-2]]) >;
%o A203510 A203510:= func< n | n eq 1 select 1 else (&*[(&*[Factorial(j) + Factorial(k): k in [1..j-1]]): j in [2..n]])/BarnesG(n+1) >;
%o A203510 [A203510(n): n in [1..13]]; // _G. C. Greubel_, Feb 24 2024
%o A203510 (SageMath)
%o A203510 def BarnesG(n): return product(factorial(j) for j in range(1,n-1))
%o A203510 def A203510(n): return product(product(factorial(j)+factorial(k) for k in range(1,j)) for j in range(1,n+1))/BarnesG(n+1)
%o A203510 [A203510(n) for n in range(1,14)] # _G. C. Greubel_, Feb 24 2024
%Y A203510 Cf. A000142, A000178, A074962, A093883, A203482, A203483.
%K A203510 nonn
%O A203510 1,2
%A A203510 _Clark Kimberling_, Jan 03 2012