A006659 Number of closed meander systems of order n+1 with n components. (Formerly M2025)

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

Offset

1,1

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.

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)

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 *)

Prog

(PARI) a(n)=2*binomial(2*n+2, n-1) \\ Charles R Greathouse IV, Dec 07 2011
(Haskell)
a006659 n = 2 * a007318' (2 * n + 2) (n - 1)
-- Reinhard Zumkeller, Jun 18 2012
(PARI) x='x+O('x^100); Vec(32/(sqrt(1-4*x)*(1+sqrt(1-4*x))^4)) \\ Altug Alkan, Oct 14 2015

Crossrefs

Cf. A002057, A001622.

Equals 2*A002694(n+1).

Cf. A005315, A005316.

A diagonal of triangle A008828.

Sequence in context: A122229 A285146 A127216 * A194771 A127221 A020522

Adjacent sequences: A006656 A006657 A006658 * A006660 A006661 A006662

Keyword

nonn,easy,nice

Author

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

Status

approved