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%I #28 Jul 07 2023 13:34:26
%S 1,2,12,110,1274,17136,255816,4124406,70549050,1264752060,23555382240,
%T 452806924752,8939481277552,180551099694400,3719061442253520,
%U 77933728043586630,1658001861319441050,35749633305661575300,780123576993991461000,17208112644166765652100
%N Number of alternating permutations on 2n+1 letters that avoid a certain pattern of length 4 (see Lewis, 2012, Appendix, for precise definition).
%C 1 together with A007724. - _Omar E. Pol_, Aug 22 2016
%H K. Gorska and K. A. Penson, <a href="http://arxiv.org/abs/1304.6008">Multidimensional Catalan and related numbers as Hausdorff moments</a>, arXiv preprint arXiv:1304.6008 [math.CO], 2013.
%H J. B. Lewis, <a href="https://dspace.mit.edu/handle/1721.1/73444">Pattern Avoidance for Alternating Permutations and Reading Words of Tableaux</a>, Ph. D. Dissertation, Department of Mathematics, MIT, 2012.
%F From _Karol A. Penson_, Aug 10 2014: (Start)
%F 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.
%F Representation of a(n) as the n-th power moment of a positive function on the segment [0,27]:
%F a(n) = int(x^n*W(x),x=0..27),n=0,1,2..., where
%F 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).
%F W(x) for x->0 has the singularity 1/sqrt(x), W(27)=0.
%F This is the solution of the Hausdorff moment problem and is unique.
%F 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).(End)
%F a(n) = A006480(n+1)/((2+n)*(1+2*n)*(3+2*n)). - _Peter Luschny_, Aug 15 2014
%F a(n) = (-1)^n*hypergeom([-2-2*n,-2*n,-2*n-1],[2,3],1). - _Peter Luschny_, Aug 29 2014
%F (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
%F a(n) ~ 3^(3*n + 7/2) / (8*Pi*n^4). - _Vaclav Kotesovec_, Jun 09 2019
%p a := n -> (-1)^n*hypergeom([-2-2*n, -2*n, -2*n-1], [2, 3], 1):
%p seq(round(evalf(a(n), 32)), n=0..20); # _Peter Luschny_, Aug 29 2014
%t Table[(3 n + 3)!/((4 (n + 1)^2 - 1) ((n + 1)!)^2 (n + 2)!), {n, 0, 20}] (* Vincenzo Librandi, Aug 30 2014 *)
%t Table[(-1)^n HypergeometricPFQ[{-2 - 2 n, -2 n, -2 n - 1}, {2, 3}, 1], {n, 0, 20}] (* _Michael De Vlieger_, Aug 22 2016 *)
%o (PARI) a(n) = (3*n+3)!/((4*(n+1)^2-1)*((n+1)!)^2*(n+2)!); \\ _Michel Marcus_, Aug 10 2014
%o (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
%Y Cf. A007724, A181197, A217799-A217830, A005789, A006480, A007724.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Oct 12 2012
%E More terms from _Alois P. Heinz_, Aug 22 2016
%E Merged with A241958 by _R. J. Mathar_, Jul 07 2023
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