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A217800
Number of alternating permutations on 2n+1 letters that avoid a certain pattern of length 4 (see Lewis, 2012, Appendix, for precise definition).
5
1, 2, 12, 110, 1274, 17136, 255816, 4124406, 70549050, 1264752060, 23555382240, 452806924752, 8939481277552, 180551099694400, 3719061442253520, 77933728043586630, 1658001861319441050, 35749633305661575300, 780123576993991461000, 17208112644166765652100
OFFSET
0,2
COMMENTS
1 together with A007724. - Omar E. Pol, Aug 22 2016
LINKS
K. Gorska and K. A. Penson, Multidimensional Catalan and related numbers as Hausdorff moments, arXiv preprint arXiv:1304.6008 [math.CO], 2013.
J. B. Lewis, Pattern Avoidance for Alternating Permutations and Reading Words of Tableaux, Ph. D. Dissertation, Department of Mathematics, MIT, 2012.
FORMULA
From Karol A. Penson, Aug 10 2014: (Start)
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).(End)
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
a(n) ~ 3^(3*n + 7/2) / (8*Pi*n^4). - Vaclav Kotesovec, Jun 09 2019
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 *)
PROG
(PARI) a(n) = (3*n+3)!/((4*(n+1)^2-1)*((n+1)!)^2*(n+2)!); \\ Michel Marcus, Aug 10 2014
(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
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Oct 12 2012
EXTENSIONS
More terms from Alois P. Heinz, Aug 22 2016
Merged with A241958 by R. J. Mathar, Jul 07 2023
STATUS
approved