A006659 Number of closed meander systems of order n+1 with n components.

2, 12, 56, 240, 990, 4004, 16016, 63648, 251940, 994840, 3922512, 15452320, 60843510, 239519700, 942871200, 3711935040, 14615744220, 57562286760, 226760523600, 893550621600, 3522078700140, 13887053160552

Offset: 1

Author

D. Ivanov, S. K. Lando and A. K. Zvonkin (zvonkin(AT)labri.u-bordeaux.fr)

Keywords

• nonn
• easy
• nice

Comments

a(n) is the total number of long interior inclines in all Dyck (n+2)-paths. An incline is a maximal subpath of like steps (all Us or all Ds); interior means it does not start or end the path; long means of length >= 2. Example: for n=1, the 5 Dyck 3-paths are shown with long interior inclines in uppercase: uuuddd, uududd, udUUdd, ududud, uuDDud and so a(1)=2. - David Callan, Jul 03 2006

a(n) is the number of corners in all parallelogram polyominoes of semiperimeter n+3. - Emeric Deutsch, Oct 09 2008

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
• M. Delest, J. P. Dubernard and I. Dutour, Parallelogram polyominoes and corners, J. Symbolic Computation, 20(1995),503-515.
• M. P. Delest, D. Gouyou-Beauchamps and B. Vauquelin, Enumeration of parallelogram polyominoes with given bond and site parameter, Graphs and Combinatorics, 3 (1987), 325-339.
• P. Di Francesco, O. Golinelli and E. Guitter, Meander, folding and arch statistics, arXiv:hep-th/9506030, 1995.
• S. K. Lando and A. K. Zvonkin, Plane and projective meanders, Séries Formelles et Combinatoire Algébrique. Laboratoire Bordelais de Recherche Informatique, Université Bordeaux I, 1991, pp. 287-303. (Annotated scanned copy)
• S. K. Lando and A. K. Zvonkin, Plane and projective meanders, Theoretical Computer Science Vol. 117, pp. 227-241, 1993.
• Simon Plouffe, Approximations of generating functions and a few conjectures, Master's Thesis UQAM 1992, arXiv:0911.4975 [math.NT], 2009.

Crossrefs

Cf. A002057, A001622.

Equals 2*A002694(n+1).

Cf. A005315, A005316.

A diagonal of triangle A008828.

Programs

Haskell

a006659 n = 2 * a007318' (2 * n + 2) (n - 1)
-- Reinhard Zumkeller, Jun 18 2012

Maple

seq(2*binomial(2*n+2,n-1),n=1..22); # Emeric Deutsch, Oct 09 2008

Mathematica

f[x_] := 32/((1 + Sqrt[1 - 4x])^4*Sqrt[1 - 4x]); CoefficientList[ Series[ f[x], {x, 0, 21}], x] (* Jean-François Alcover, Dec 07 2011 *)
CoefficientList[Series[4*Exp[2x](BesselI[1,2*x]+ x(x-1)(BesselI[0,2x]+BesselI[1,2x]))/x^2,{x,0,22}],x]Table[n!,{n,0,22}] (* Stefano Spezia, May 10 2022 *)

PARI

a(n)=2*binomial(2*n+2,n-1) \\ Charles R Greathouse IV, Dec 07 2011

PARI

x='x+O('x^100); Vec(32/(sqrt(1-4*x)*(1+sqrt(1-4*x))^4)) \\ Altug Alkan, Oct 14 2015

Formula

G.f.: 32/(sqrt(1-4x)*(1+sqrt(1-4x))^4).

a(n) = (n+1) * A002057(n). - Ralf Stephan, Aug 31 2003

a(n) = 2*binomial(2n+2, n-1). - Emeric Deutsch, Oct 09 2008

a(n) = {(-56 - 30*n - 4*n^2)*a(n+1) + (8*n+12+n^2)*a(n+2), a(0)=2, a(1)=12}. - Simon Plouffe (master's thesis, 1992)

a(n) ~ 2^(2*n+3)/sqrt(n*Pi). - Charles R Greathouse IV, Dec 07 2011

E.g.f.: 4*exp(2*x)*(I_1(2*x) + x*(x - 1)*(I_0(2*x) + I_1(2*x)))/x^2, where I_n(x) is the modified Bessel function of the first kind. - Stefano Spezia, May 09 2022

From Amiram Eldar, May 15 2022: (Start)

Sum_{n>=1} 1/a(n) = 23/12 - 13*Pi/(18*sqrt(3)).

Sum_{n>=1} (-1)^(n+1)/a(n) = 53*log(phi)/(5*sqrt(5)) - 37/20, where phi is the golden ratio (A001622). (End)