A241958 Duplicate of A217800.

1, 2, 12, 110, 1274, 17136, 255816, 4124406, 70549050, 1264752060, 23555382240, 452806924752, 8939481277552, 180551099694400, 3719061442253520, 77933728043586630, 1658001861319441050, 35749633305661575300, 780123576993991461000, 17208112644166765652100, 383292388823513983713900

Offset: 0

Author

Karol A. Penson, Aug 10 2014

Keywords

• dead

Comments

This is a duplicate of A217800 or of A007724. - Alois P. Heinz, Aug 22 2016

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• K. Gorska and K. A. Penson, Multidimensional Catalan and related numbers as Hausdorff moments, arXiv preprint arXiv:1304.6008, 2013

Programs

Magma

[Factorial(3*n+3)/((4*(n+1)^2-1)*Factorial((n+1))^2*Factorial(n+ 2)): n in [0..20]]; // Vincenzo Librandi, Aug 30 2014

Maple

a := n -> (-1)^n*hypergeom([-2-2*n,-2*n,-2*n-1],[2, 3],1):
seq(round(evalf(a(n),32)),n=0..20); # Peter Luschny, Aug 29 2014

Mathematica

Table[(3 n + 3)!/((4 (n + 1)^2 - 1) ((n + 1)!)^2 (n + 2)!), {n, 0, 20}] (* Vincenzo Librandi, Aug 30 2014 *)
Table[(-1)^n HypergeometricPFQ[{-2 - 2 n, -2 n, -2 n - 1}, {2, 3},
1], {n, 0, 20}] (* Michael De Vlieger, Aug 22 2016 *)

PARI

a(n) = (3*n+3)!/((4*(n+1)^2-1)*((n+1)!)^2*(n+2)!); \\ Michel Marcus, Aug 10 2014

Formula

O.g.f.(in Maple notation): hypergeom([1/2, 1, 4/3, 5/3], [2, 5/2, 3], 27*z);

a(n) ~ (1/93312)*sqrt(3)*27^n*(314928*n^4-1644624*n^3+5545260*n^2 -15387660*n+38310503)/(Pi*n^8), for n -> infinity.

Representation of a(n) as the n-th power moment of a positive function on the segment [0,27]:

a(n) = int(x^n*W(x),x=0..27),n=0,1,2..., where

W(x) = 1/(Pi*sqrt(x))+sqrt(x)/Pi-(9/20)*sqrt(3)*2^(1/3)* hypergeom([-2/3, -1/6, 1/3], [2/3, 11/6], (1/27)*x)*x^(1/3)/ (sqrt(Pi)*Gamma(5/6)*Gamma(2/3))-(27/56)*2^(2/3)*Gamma(5/6) *Gamma(2/3)*hypergeom([-1/3, 1/6, 2/3], [4/3, 13/6], (1/27)*x)* x^(2/3)/Pi^(5/2).

W(x) for x->0 has the singularity 1/sqrt(x), W(27)=0.

This is the solution of the Hausdorff moment problem and is unique.

a(n) = (1/2)*(n+3)!/((4*(n+1)^2-1)*(n+1)!)*A005789(n), where A005789(n) are the three-dimensional Catalan numbers (see the Gorska and Penson link).

a(n) = A006480(n+1)/((2+n)*(1+2*n)*(3+2*n)). - Peter Luschny, Aug 15 2014

a(n) = (-1)^n*hypergeom([-2-2*n,-2*n,-2*n-1],[2,3],1). - Peter Luschny, Aug 29 2014

(2*n+3)*(n+2)*(n+1)*a(n) -3*(3*n+2)*(2*n-1)*(3*n+1)*a(n-1)=0. - R. J. Mathar, Jun 14 2016