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A203513 a(n) = A203312(n+1)/A203312(n).

%I A203513 #27 Feb 23 2024 02:31:01
%S A203513 3,49,2028,159201,20342448,3850078401,1012487793408,353293863908769,
%T A203513 157973407966483200,88087149666575064369,59928191584204259377152,
%U A203513 48860028872008706126041281
%N A203513 a(n) = A203312(n+1)/A203312(n).
%C A203513 See A093883 for a discussion and guide to related sequences.
%H A203513 G. C. Greubel, <a href="/A203513/b203513.txt">Table of n, a(n) for n = 1..215</a>
%F A203513 a(n) ~ exp((n+1)*Pi/(sqrt(3)) - 2*n) * n^(2*n). - _Vaclav Kotesovec_, Sep 07 2023
%t A203513 f[j_] := j; z = 12;
%t A203513 v[n_] := Product[Product[f[j]^2 - f[j] f[k] + f[k]^2,
%t A203513 {j, 1, k - 1}], {k, 2, n}]
%t A203513 Table[v[n], {n, 1, z}]          (* A203312 *)
%t A203513 Table[v[n + 1]/v[n], {n, 1, z}] (* A203513 *)
%t A203513 Table[Product[k^2 - k*(n+1) + (n+1)^2, {k, 1, n}], {n, 1, 15}] (* _Vaclav Kotesovec_, Sep 07 2023 *)
%o A203513 (Python)
%o A203513 from operator import mul
%o A203513 from functools import reduce
%o A203513 def v(n): return 1 if n==1 else reduce(mul, [reduce(mul, [j**2 - j*k + k**2 for j in range(1, k)]) for k in range(2, n + 1)])
%o A203513 print([v(n + 1)//v(n) for n in range(1, 13)]) # _Indranil Ghosh_, Jul 26 2017
%o A203513 (Magma) [(&*[(n+1)*(n-j+1) +j^2: j in [1..n]]): n in [1..30]]; // _G. C. Greubel_, Feb 23 2024
%o A203513 (SageMath)
%o A203513 def A203513(n): return product((n+1)*(n-j+1) +j^2 for j in range(1, n+1))
%o A203513 [A203513(n) for n in range(1,31)] # _G. C. Greubel_, Feb 23 2024
%Y A203513 Cf. A093883, A203312.
%K A203513 nonn
%O A203513 1,1
%A A203513 _Clark Kimberling_, Jan 04 2012