A383874 a(n) = (3*n+1)!*(3*n)!/((2*n)!*((n+1)!)^2).
1, 18, 4200, 3175200, 5137292160, 14544244915200, 64008493310361600, 405192226643043840000, 3493057136053143859200000, 39378260464472988708249600000, 562659674639968187756457984000000, 9940535265182157971578474463232000000, 212816707229761791940688046273331200000000
Offset: 0
Keywords
Links
- Paolo Xausa, Table of n, a(n) for n = 0..150
Programs
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Mathematica
A383874[n_] := (3*n+1)!*(3*n)!/((2*n)!*((n+1)!)^2); Array[A383874, 15, 0] (* Paolo Xausa, May 26 2025 *)
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PARI
a(n) = (3*n+1)!*(3*n)!/((2*n)!*((n+1)!)^2); \\ Michel Marcus, May 22 2025
Formula
O.g.f.: hypergeom([1/3, 2/3, 2/3, 1, 1, 4/3], [1/2, 2, 2], (729*x)/4).
E.g.f.: hypergeom([1/3, 2/3, 2/3, 1, 1, 4/3], [1/2, 2, 2, 1], (729*x)/4).
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.
W(x) is the solution of the Stieltjes moment problem and it may be non-unique.
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