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%I M2025 #78 Jun 17 2022 03:23:28
%S 2,12,56,240,990,4004,16016,63648,251940,994840,3922512,15452320,
%T 60843510,239519700,942871200,3711935040,14615744220,57562286760,
%U 226760523600,893550621600,3522078700140,13887053160552
%N Number of closed meander systems of order n+1 with n components.
%C 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
%C a(n) is the number of corners in all parallelogram polyominoes of semiperimeter n+3. - _Emeric Deutsch_, Oct 09 2008
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Reinhard Zumkeller, <a href="/A006659/b006659.txt">Table of n, a(n) for n = 1..1000</a>
%H M. Delest, J. P. Dubernard and I. Dutour, <a href="http://dx.doi.org/10.1006/jsco.1995.1062">Parallelogram polyominoes and corners</a>, J. Symbolic Computation, 20(1995),503-515.
%H M. P. Delest, D. Gouyou-Beauchamps and B. Vauquelin, <a href="http://dx.doi.org/10.1007/BF01788555">Enumeration of parallelogram polyominoes with given bond and site parameter</a>, Graphs and Combinatorics, 3 (1987), 325-339.
%H P. Di Francesco, O. Golinelli and E. Guitter, <a href="http://arXiv.org/abs/hep-th/9506030">Meander, folding and arch statistics</a>, arXiv:hep-th/9506030, 1995.
%H S. K. Lando and A. K. Zvonkin, <a href="/A005316/a005316_1.pdf">Plane and projective meanders</a>, Séries Formelles et Combinatoire Algébrique. Laboratoire Bordelais de Recherche Informatique, Université Bordeaux I, 1991, pp. 287-303. (Annotated scanned copy)
%H S. K. Lando and A. K. Zvonkin, <a href="http://dx.doi.org/10.1016/0304-3975(93)90316-L">Plane and projective meanders</a>, Theoretical Computer Science Vol. 117, pp. 227-241, 1993.
%H Simon Plouffe, <a href="http://arxiv.org/abs/0911.4975">Approximations of generating functions and a few conjectures</a>, Master's Thesis UQAM 1992, arXiv:0911.4975 [math.NT], 2009.
%F G.f.: 32/(sqrt(1-4x)*(1+sqrt(1-4x))^4).
%F a(n) = (n+1) * A002057(n). - _Ralf Stephan_, Aug 31 2003
%F a(n) = 2*binomial(2n+2, n-1). - _Emeric Deutsch_, Oct 09 2008
%F 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)
%F a(n) ~ 2^(2*n+3)/sqrt(n*Pi). - _Charles R Greathouse IV_, Dec 07 2011
%F 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
%F From _Amiram Eldar_, May 15 2022: (Start)
%F Sum_{n>=1} 1/a(n) = 23/12 - 13*Pi/(18*sqrt(3)).
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 53*log(phi)/(5*sqrt(5)) - 37/20, where phi is the golden ratio (A001622). (End)
%p seq(2*binomial(2*n+2,n-1),n=1..22); # _Emeric Deutsch_, Oct 09 2008
%t 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 *)
%t 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 *)
%o (PARI) a(n)=2*binomial(2*n+2,n-1) \\ _Charles R Greathouse IV_, Dec 07 2011
%o (Haskell)
%o a006659 n = 2 * a007318' (2 * n + 2) (n - 1)
%o -- _Reinhard Zumkeller_, Jun 18 2012
%o (PARI) x='x+O('x^100); Vec(32/(sqrt(1-4*x)*(1+sqrt(1-4*x))^4)) \\ _Altug Alkan_, Oct 14 2015
%Y Cf. A002057, A001622.
%Y Equals 2*A002694(n+1).
%Y Cf. A005315, A005316.
%Y A diagonal of triangle A008828.
%K nonn,easy,nice
%O 1,1
%A D. Ivanov, S. K. Lando and A. K. Zvonkin (zvonkin(AT)labri.u-bordeaux.fr)
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