A008828 -id:A008828 - cp's OEIS frontend

A046690 Duplicate of A008828.

1, 2, 2, 8, 12, 5, 42, 84, 56, 14, 262, 640, 580, 240, 42, 1828, 5236, 5894, 3344, 990

Offset: 1

dead

A001246 Squares of Catalan numbers.

1, 1, 4, 25, 196, 1764, 17424, 184041, 2044900, 23639044, 282105616, 3455793796, 43268992144, 551900410000, 7152629313600, 93990019574025, 1250164827828900, 16807771574144100, 228138727737690000, 3123219182728976100, 43087676888260976400, 598598221893939680400, 8369059450146650049600

Offset: 0

N. J. A. Sloane

Also multi-component meanders.
Also, number of walks within N^2 (the first quadrant of Z^2) starting and ending at (0,0) and consisting of 2 n steps taken from {(-1, -1), (-1, 1), (1, -1), (1, 1)}. [Evans and Pugh show that this is the same sequence.] - N. J. A. Sloane, Jul 04 2014
This is probably the diagonal of A209805. In this case a(n) = number of non-crossing partitions up to rotation of [2n+1] into n+1 blocks. See "Partition related number triangles" in Links section. - Tilman Piesk, Apr 09 2012
a(n) is also the number of regular cover graphs of lattice quotients of essential lattice congruences of the weak order on the symmetric group S_{n+1}. See Table 1 in the Hoang/Mütze reference in the Links section. - Torsten Muetze, Nov 28 2019

T. D. Noe, Table of n, a(n) for n = 0..100
Andrei Asinowski, Cyril Banderier, and Sarah J. Selkirk, From Kreweras to Gessel: A walk through patterns in the quarter plane, Séminaire Lotharingien de Combinatoire, Proc. 35th Conf. Formal Power Series and Alg. Comb. (Davis, 2023) Vol. 89B, Art. #30.
Cyril Banderier, Markus Kuba, Stephan Wagner, and Michael Wallner, Composition schemes: q-enumerations and phase transitions, arXiv:2311.17226 [math.CO], 2023. See p. 6.
Mireille Bousquet-Mélou and Marni Mishna, Walks with small steps in the quarter plane, arXiv:0810.4387 [math.CO], 2008-2009.
P. Di Francesco, O. Golinelli, and E. Guitter, Meander, folding and arch statistics, arXiv:hep-th/9506030, 1995.
David E. Evans and Mathew Pugh, Spectral measures associated to rank two Lie groups and finite subgroups of GL(2,Z), arXiv preprint arXiv:1404.1877 [math.OA], 2014-2015.
O. Guibert, Stack words, standard Young tableaux, permutations with forbidden subsequences and planar maps, Discr. Math., Vol. 210, No. 1-3 (2000), pp. 71-85.
Hung Phuc Hoang and Torsten Mütze, Combinatorial generation via permutation languages. II. Lattice congruences, arXiv:1911.12078 [math.CO], 2019.
Amya Luo, Pattern Avoidance in Nonnesting Permutations, Undergraduate Thesis, Dartmouth College (2024). See p. 11.
Tilman Piesk, Partition related number triangles. (Wikiversity article)
Wikipedia, Ramanujan-Sato series.

Cf. A000108, A000356, A000891, A186264.
Row sums of triangle A008828.
Probably diagonal of A209805.

List([0..25],n->(Binomial(2*n,n)/(n+1))^2); # Muniru A Asiru, Mar 28 2018

seq((binomial(2*n,n)/(1+n))^2, n=0..18); # Zerinvary Lajos, Jun 18 2007

aux[i_Integer, j_Integer, n_Integer] := Which[Min[i, j, n] < 0 || Max[i, j] > n, 0, n == 0, KroneckerDelta[i, j, n], True, aux[i, j, n] = aux[-1 + i, -1 + j, -1 + n] + aux[-1 + i, 1 + j, -1 + n] + aux[1 + i, -1 + j, -1 + n] + aux[1 + i, 1 + j, -1 + n]]; Table[aux[0, 0, 2 n], {n, 0, 25}] (* Manuel Kauers, Nov 18 2008 *)
CatalanNumber[Range[0,30]]^2  (* Harvey P. Dale, Apr 26 2011 *)
a[ n_] := If[ n == -1, 0, CatalanNumber[ n]^2] (* Michael Somos, Jul 11 2011 *)
a[ n_] := SeriesCoefficient[ (2 EllipticE[ 16 x] - (1 - 16 x) EllipticK[ 16 x] - Pi/2) / ( 2 Pi x), {x, 0, n}] (* Michael Somos, Jul 11 2011 *)
a[ n_] := If[ n < 0, 0, (2 n)! SeriesCoefficient[ HypergeometricPFQ[ {1/2}, {2, 2}, 4 x^2], {x, 0, 2 n}]] (* Michael Somos, Jul 11 2011 *)

combinat::dyckWords::count(n)^2 $ n = 0..18 // Zerinvary Lajos, Feb 15 2007

a(n)=(binomial(2*n,n)/(n+1))^2 \\ Charles R Greathouse IV, Jul 16 2011

[catalan_number(i)^2 for i in range(0,19)] # Zerinvary Lajos, May 17 2009

G.f.: -1/(4*x)+1/2*(16*x-1)/x * EllipticK(4*x^(1/2))/Pi + 1/x*EllipticE(4*x^(1/2))/Pi. - Vladeta Jovovic, Oct 12 2003
G.f.: 3F2( (1, 1/2, 1/2); (2, 2); 16x) = (-1 + 2F1( (-1/2, -1/2); (1); 16x))/(4*x) - Olivier Gérard, Feb 16 2011
E.g.f.: hypergeom([1/2], [2, 2], 4*x^2) = 2*BesselI(0, 2*x)^2-BesselI(0, 2*x)*BesselI(1, 2*x)/x-2*BesselI(1, 2*x)^2. - Vladeta Jovovic, Jun 04 2005
D-finite with recurrence (n+1)^2*a(n) -4*(2*n-1)^2*a(n-1)=0. - R. J. Mathar, Jan 04 2013
From Ilya Gutkovskiy, Mar 23 2017: (Start)
a(n) ~ 16^n/(Pi*n^3).
Sum_{n>=0} 1/a(n) = 3F2(1,2,2; 1/2,1/2; 1/16) = 2.295732295098655... (End)
Sum {n>=0} a(n)*(n+1)/16^n = 4/Pi. This is a kind of Ramanujan-Sato series. - Ralf Steiner, Mar 23 2017
From Peter Bala, Mar 28 2018: (Start)
a(n) = 1/(2*n + 1)*f(2*n)/(f(n)*f(n)), where f(n) = n!*(n+1)!. Cf. Catalan(n) = 1/(n + 1)*(2*n)!/(n!*n!).
a(n) = 1/(2*n + 1)*A000891(n).
a(n) = (n + 2)/(2*n + 1)*A000356(n).
a(n) = (n + 2)/3*A186264(n-1). (End)
From Amiram Eldar, Mar 27 2022: (Start)
a(n) = A000108(n)^2.
Sum_{n>=0} a(n)/16^n = 16/Pi - 4. (End)

As a result of the work of Evans and Pugh, it was possible to merge A151342 with this sequence. - N. J. A. Sloane, Jul 04 2014

A005315 Closed meandric numbers (or meanders): number of ways a loop can cross a road 2n times.

Original entry on oeis.org

1, 1, 2, 8, 42, 262, 1828, 13820, 110954, 933458, 8152860, 73424650, 678390116, 6405031050, 61606881612, 602188541928, 5969806669034, 59923200729046, 608188709574124, 6234277838531806, 64477712119584604, 672265814872772972, 7060941974458061392

Offset: 0

N. J. A. Sloane, J. A. Reeds (reeds(AT)idaccr.org)

There is a 1-to-1 correspondence between loops crossing a road 2n times and lines crossing a road 2n-1 times.

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.
S. K. Lando and A. K. Zvonkin, Meanders, Selecta Mathematica Sovietica, Vol. 11, Number 2, pp. 117-144, 1992.
A. Phillips, Simple Alternating Transit Mazes, preprint. Abridged version appeared as "La topologia dei labirinti," in M. Emmer, editor, L'Occhio di Horus: Itinerari nell'Imaginario Matematico. Istituto della Enciclopedia Italia, Rome, 1989, pp. 57-67.
V. R. Pratt, personal communication.
J. A. Reeds and L. A. Shepp, An upper bound on the meander constant, preprint, May 25, 1999. [Obtains upper bound of 13.01]
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
For additional references, see A005316.

Andrew Howroyd, Table of n, a(n) for n = 0..28 (first 24 terms from Iwan Jensen)
Oswin Aichholzer, Carlos Alegría Galicia, Irene Parada, Alexander Pilz, Javier Tejel, Csaba D. Tóth, Jorge Urrutia, and Birgit Vogtenhuber, Hamiltonian meander paths and cycles on bichromatic point sets, XVIII Spanish Meeting on Computational Geometry (Girona, 2019).
V. I. Arnol'd, A branched covering of CP^2->S^4, hyperbolicity and projective topology [ Russian ], Sibir. Mat. Zhurn., 29 (No. 2, 1988), 36-47 = Siberian Math. J., 29 (1988), 717-725.
Roland Bacher, Meander algebras
David Bevan, Random Closed Meanders - _David Bevan_, Jun 25 2010
Alexander E. Black, Kevin Liu, Alex Mcdonough, Garrett Nelson, Michael C. Wigal, Mei Yin, and Youngho Yoo, Sampling planar tanglegrams and pairs of disjoint triangulations, arXiv:2304.05318 [math.CO], 2023.
B. Bobier and J. Sawada, A fast algorithm to generate open meandric systems and meanders, Transactions on Algorithms, Vol. 6 No. 2 (2010) article #42, 12 pages.
P. Di Francesco, O. Golinelli and E. Guitter, Meander, folding and arch statistics, arXiv:hep-th/9506030, 1995.
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, 2009; see page 525.
Reinhard O. W. Franz, and Berton A. Earnshaw, A constructive enumeration of meanders, Ann. Comb. 6 (2002), no. 1, 7-17.
Erich Friedman, Illustration of initial terms
Motohisa Fukuda, Ion Nechita, Enumerating meandric systems with large number of components, arXiv preprint arXiv:1609.02756 [math.CO], 2016.
Iwan Jensen, Home page
Iwan Jensen, Closed meanders, a(n) for n = 1..24
Iwan Jensen, Enumeration of plane meanders, arXiv:cond-mat/9910313 [cond-mat.stat-mech], 1999.
Iwan Jensen, A transfer matrix approach to the enumeration of plane meanders, J. Phys. A 33, 5953-5963 (2000).
Iwan Jensen and A. J. Guttmann, Critical exponents of plane meanders J. Phys. A 33, L187-L192 (2000).
Michael La Croix, Approaches to the Enumerative Theory of Meanders, 2003.
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.
A. Panayotopoulos, On Meandric Colliers, Mathematics in Computer Science, (2018).
A. Panayotopoulos and P. Tsikouras, Meanders and Motzkin Words, J. Integer Seqs., Vol. 7, 2004.
A. Phillips, Mazes
A. Phillips, Simple, Alternating, Transit Mazes
J. A. Reeds, D. E. Knuth, & N. J. A. Sloane, Email Correspondence
J. Reeds, L. Shepp, & D. McIlroy, Numerical bounds for the Arnol'd "meander" constant, Preprint.
M. A. Sainte-Laguë, Les Réseaux (ou Graphes), Mémorial des Sciences Mathématiques, Fasc. 18, Gauthier-Villars, Paris, 1926, p. 47.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).

These are the odd-numbered terms of A005316. Cf. A077054. For nonisomorphic solutions, see A077460.

A column of triangle A008828.

A005316 = Cases[Import["https://oeis.org/A005316/b005316.txt", "Table"], {, }][[All, 2]];
a[n_] := If[n == 0, 1, A005316[[2n]]];
a /@ Range[0, 28] (* Jean-François Alcover, Sep 25 2019 *)

a(n) = A005316(2n-1) for n>0.

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

Original entry on oeis.org

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

Offset: 1

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

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

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

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.

Cf. A002057, A001622.
Equals 2*A002694(n+1).
Cf. A005315, A005316.
A diagonal of triangle A008828.

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

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

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 *)

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

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

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)

A259974 Irregular triangle read by rows: T(n,k) = number of meanders with n bridges in which the first bridge is bridge k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 2, 3, 8, 3, 3, 14, 7, 7, 14, 42, 14, 11, 14, 81, 36, 28, 36, 81, 262, 81, 57, 57, 81, 538, 221, 155, 155, 221, 538, 1828, 538, 353, 316, 353, 538, 3926, 1530, 1003, 902, 1003, 1530, 3926, 13820, 3926, 2458, 2053, 2053, 2458, 3926, 30694, 11510, 7214, 6059, 6059, 7214, 11510, 30694, 110954, 30694, 18575, 14810, 13827, 14810, 18575, 30694

Offset: 1

N. J. A. Sloane, Jul 12 2015

			Triangle begins:
1,
1,
1,1,
2,1,
3,2,3,
8,3,3,
14,7,7,14,
42,14,11,14,
81,36,28,36,81,
262,81,57,57,81,
538,221,155,155,221,538,
...

Lando, S. K. and Zvonkin, A. K. Plane and projective meanders. Conference on Formal Power Series and Algebraic Combinatorics (Bordeaux, 1991).
Lando, S. K. and Zvonkin, A. K. Meanders. In Selected translations. Selecta Math. Soviet. 11 (1992), no. 2, 117-144.

Andrew Howroyd, Table of n, a(n) for n = 1..420
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 (1993), no. 1-2, 227-241.

Diagonals are A005316, A006660, A006661, A006662. Cf. A008828.

T(12,k)-T(40,k) from Andrew Howroyd, Dec 15 2015

A006657 Number of closed meanders with 2 components and 2n bridges.

Original entry on oeis.org

2, 12, 84, 640, 5236, 45164, 406012, 3772008, 35994184, 351173328, 3490681428, 35253449296, 360946635312, 3739935635756, 39159200588780, 413836299216608, 4409705753032648, 47337525317450816, 511563350415103008

Offset: 2

N. J. A. Sloane, Simon Plouffe

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

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)

Cf. A005315.

A column of triangle A008828.

A008828 = Import["https://oeis.org/A008828/b008828.txt", "Table"][[All, 2]];
a[n_] := A008828[[(n^2 - n + 4)/2]];
a /@ Range[2, 20] (* Jean-François Alcover, Sep 25 2019 *)

a(13)-a(20) from Andrew Howroyd, Nov 22 2015

A007746 Number of ways for n-3 nonintersecting loops to cross a line 2n times.

Original entry on oeis.org

42, 640, 5894, 42840, 271240, 1569984, 8536890, 44346456, 222516030, 1086685600, 5193298110, 24384586200, 112831907760, 515709552000, 2332549535400, 10455495457248, 46500885666900, 205401168733824, 901819865269180, 3938266773556720, 17116175702216624

Offset: 4

Philippe Di Francesco (philippe(AT)amoco.saclay.cea.fr)

nonn

Andrew Howroyd, Table of n, a(n) for n = 4..50
P. Di Francesco, O. Golinelli and E. Guitter, Meanders: a direct enumeration approach, Nucl. Phys. B 482 [ FS ] (1996) 497-535.
S. K. Lando and A. K. Zvonkin, Plane and projective meanders, Theoretical Computer Science Vol. 117, pp. 227-241, 1993.

A diagonal of triangle A008828.

[4*Factorial(2*n)/(3*Factorial(n-4)*Factorial(n+6))* (n^4+20*n^3+107*n^2-107*n+15): n in [4..25]]; // Vincenzo Librandi, Nov 23 2015

Table[4 (2 n)!/(3 (n - 4)! (n+6)!) (n^4 + 20 n^3 + 107 n^2 - 107 n + 15), {n, 4, 30}] (* Vincenzo Librandi, Nov 23 2015 *)

a(n) = 4 * (2*n)! * (n^4+20*n^3+107*n^2-107*n+15) / ( 3*(n-4)! * (n+6)! ).

A380368 Triangle read by rows: T(n,k) is the number of closed forest meander systems with 2n crossings and k components.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 8, 6, 1, 0, 42, 42, 12, 1, 0, 262, 320, 130, 20, 1, 0, 1828, 2618, 1360, 310, 30, 1, 0, 13820, 22582, 14196, 4270, 630, 42, 1, 0, 110954, 203006, 149024, 55524, 11060, 1148, 56, 1, 0, 933458, 1886004, 1577712, 698952, 175560, 25032, 1932, 72, 1

Offset: 0

Andrew Howroyd, Jan 31 2025

A forest meander system is a meander system that does not have any components which are entirely enclosed by another. An equivalent condition is that all components have their least point at an odd index (if points are numbered from 1). The greatest point will then be at an even index.

Exactly half of all meander systems with two components are forest meander systems. This is because when the meander's permutation is rotated one step at a time, one meander will be enclosed in the other on every second step.

			Triangle begins:
  1;
  0,     1;
  0,     2,     1;
  0,     8,     6,     1;
  0,    42,    42,    12,    1;
  0,   262,   320,   130,   20,   1;
  0,  1828,  2618,  1360,  310,  30,  1;
  0, 13820, 22582, 14196, 4270, 630, 42, 1;
  ...
The T(3,2) = 6 forest meander systems are the following and their reflections.
       ______
      / ____ \                 ___
     / /    \ \               /   \
 .. / /. /\ .\ \ ..   and .. / / \ \ . /\ ..
    \/   \/   \/             \/   \/   \/
        (2)                     (4)
.
There are also 6 systems that are not forest meander systems:
      ____                    ______
     / __ \                  /      \
 .. / /  \ \ ..      and .. / /\  /\ \ ..
    \ \/\/ /                \ \/ /  \/
     \____/                  \__/
       (2)                     (4)

Andrew Howroyd, Table of n, a(n) for n = 0..230 (rows 0..20)
Roland Bacher, Meander algebras.

Row sums are A060148.
Column k=1 is A005315.
Column k=2 is half of A006657.
Main diagonal is A000012.
Second diagonal is A002378.
Cf. A008828 (all meander systems), A060174, A060198.

A006658 Closed meanders with 3 components and 2n bridges.

Original entry on oeis.org

5, 56, 580, 5894, 60312, 624240, 6540510, 69323910, 742518832, 8028001566, 87526544560, 961412790002, 10630964761766, 118257400015312, 1322564193698320, 14863191405246888, 167771227744292160, 1901345329566422790

Offset: 3

N. J. A. Sloane

nonn

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

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)

Cf. A005315, A006657.

A column of triangle A008828.

A008828 = Import["https://oeis.org/A008828/b008828.txt", "Table"][[All, 2]];
a[n_] := A008828[[(n^2 - n + 6)/2]];
a /@ Range[3, 20] (* Jean-François Alcover, Sep 25 2019 *)

a(13)-a(20) from Andrew Howroyd, Nov 22 2015

A368054 Irregular triangle read by rows: T(n,k) is the number of k-crossing partitions on 2n nodes, where all partition terms alternate in parity, counted up to reflection.

Original entry on oeis.org

1, 1, 3, 0, 1, 14, 0, 8, 10, 2, 2, 81, 0, 59, 162, 70, 66, 82, 22, 19, 6, 7, 0, 2, 538, 0, 454, 1952, 1229, 1208, 2516, 1803, 1181, 1148, 998, 478, 370, 279, 125, 76, 26, 13, 3, 3, 3926, 0, 3658, 21608, 17083, 17811, 48542, 51306, 40081, 51660, 59023, 42327

Offset: 0

John Tyler Rascoe, Dec 09 2023

The 0-crossing partitions counted in A005316 all have terms that alternate in parity. Also, for an even number of nodes the partitions 1432 and 2341 count the same meandric path. This triangle aims to reduce the total number of k-crossing partitions considered from (2*n)! to (n!)^2, see Irwin link.

			Triangle begins:
       k=0  1   2    3   4   5   6   7   8   9  10  11  12
  n=0:   1;
  n=1:   1;
  n=2:   3, 0,  1;
  n=3:  14, 0,  8,  10,  2,  2;
  n=4:  81, 0, 59, 162, 70, 66, 82, 22, 19,  6,  7,  0,  2;
  ...
Row n = 3 counts the following k-crossing partitions.
T(3,0) = 14:   T(3,2) = 8:    T(3,3) = 10:   T(3,4) = 2:    T(3,5) = 2:
(1,2,3,4,5,6)  (3,4,1,6,5,2)  (1,2,5,6,3,4)  (3,2,5,6,1,4)  (3,6,1,4,5,2)
(1,2,3,6,5,4)  (3,4,5,6,1,2)  (1,4,3,6,5,2)  (3,6,1,2,5,4)  (5,2,3,6,1,4)
(1,2,5,4,3,6)  (3,6,5,4,1,2)  (1,4,5,2,3,6)
(1,4,3,2,5,6)  (5,2,1,6,3,4)  (1,6,3,2,5,4)
(1,4,5,6,3,2)  (5,4,3,6,1,2)  (3,2,5,4,1,6)
(1,6,3,4,5,2)  (5,6,1,2,3,4)  (3,4,1,2,5,6)
(1,6,5,2,3,4)  (5,6,1,4,3,2)  (3,6,5,2,1,4)
(1,6,5,4,3,2)  (5,6,3,2,1,4)  (5,2,1,4,3,6)
(3,2,1,4,5,6)                 (5,4,1,6,3,2)
(3,2,1,6,5,4)                 (5,6,3,4,1,2)
(3,4,5,2,1,6)
(5,2,3,4,1,6)
(5,4,1,2,3,6)
(5,4,3,2,1,6)

Benedict Irwin, On the Number of k-Crossing Partitions, Univ. of Cambridge (2021).
John Tyler Rascoe, Python program.

Cf. A077054 (column k=0), A001044 (row sums).

Cf. A005316, A008828, A287220.

Python
```
# see linked program
```

cp's OEIS Frontend

A046690 Duplicate of A008828.

Views

Author

Keywords

A001246 Squares of Catalan numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Maple

Mathematica

MuPAD

PARI

Sage

Formula

Extensions

A005315 Closed meandric numbers (or meanders): number of ways a loop can cross a road 2n times.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

Formula

A259974 Irregular triangle read by rows: T(n,k) = number of meanders with n bridges in which the first bridge is bridge k.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Extensions

A006657 Number of closed meanders with 2 components and 2n bridges.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

Extensions

A007746 Number of ways for n-3 nonintersecting loops to cross a line 2n times.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A380368 Triangle read by rows: T(n,k) is the number of closed forest meander systems with 2n crossings and k components.

Views

Author

Keywords