This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A383874 #124 May 26 2025 11:27:43
%S A383874 1,18,4200,3175200,5137292160,14544244915200,64008493310361600,
%T A383874 405192226643043840000,3493057136053143859200000,
%U A383874 39378260464472988708249600000,562659674639968187756457984000000,9940535265182157971578474463232000000,212816707229761791940688046273331200000000
%N A383874 a(n) = (3*n+1)!*(3*n)!/((2*n)!*((n+1)!)^2).
%H A383874 Paolo Xausa, <a href="/A383874/b383874.txt">Table of n, a(n) for n = 0..150</a>
%F A383874 O.g.f.: hypergeom([1/3, 2/3, 2/3, 1, 1, 4/3], [1/2, 2, 2], (729*x)/4).
%F A383874 E.g.f.: hypergeom([1/3, 2/3, 2/3, 1, 1, 4/3], [1/2, 2, 2, 1], (729*x)/4).
%F A383874 a(n) = Integral_{x>=0} x^n*W(x)*dx, n>=0, with W(x) = MeijerG([[],[-1/2,1,1]],[[0,-1/3,-1/3,1/3,-2/3],[]],4*x/729)/(81*Pi^(3/2)), where MeijerG is the Meijer G - function. Apparently W(x) cannot be represented by any other simpler functions. W(x) is a positive function on (0,oo), is singular at x = 0 and goes monotonically to zero as x -> oo. Thus a(n) is a positive definite sequence.
%F A383874 W(x) is the solution of the Stieltjes moment problem and it may be non-unique.
%F A383874 a(n) ~ 3^(6*n+2) * n^(2*n - 3/2) / (sqrt(Pi) * 2^(2*n+1) * exp(2*n)). - _Vaclav Kotesovec_, May 24 2025
%t A383874 A383874[n_] := (3*n+1)!*(3*n)!/((2*n)!*((n+1)!)^2);
%t A383874 Array[A383874, 15, 0] (* _Paolo Xausa_, May 26 2025 *)
%o A383874 (PARI) a(n) = (3*n+1)!*(3*n)!/((2*n)!*((n+1)!)^2); \\ _Michel Marcus_, May 22 2025
%Y A383874 Cf. A166149, A166771, A166494, A166384, A166750, A271049, A064352.
%K A383874 nonn
%O A383874 0,2
%A A383874 _Karol A. Penson_, May 22 2025