A005700 a(n) = C(n)*C(n+2) - C(n+1)^2 where C() are the Catalan numbers A000108.

1, 1, 3, 14, 84, 594, 4719, 40898, 379236, 3711916, 37975756, 403127256, 4415203280, 49671036900, 571947380775, 6721316278650, 80419959684900, 977737404590100, 12058761323277900, 150656212896017400, 1904342169333848400, 24328661192286773400, 313839729380499376860

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• walk
• easy

Comments

The old name was: Number of closed walks of 2n unit steps north, east, south, or west starting and ending at the origin and confined to the first octant.

Image of Catalan numbers (A000108) under "little Hankel" transform that sends [c_0, c_1, ...] to [d_0, d_1, ...] where d_n = c_n^2 - c_{n+1}*c_{n-1}.

The Niederhausen reference counts various classes of first octant paths by number of contacts with the line y=x. - David Callan, Sep 18 2007

In Sloane and Plouffe (1995) this was incorrectly described as "Dyck paths".

Also matchings avoiding a certain pattern (see J. Bloom and S. Elizalde). - N. J. A. Sloane, Jan 02 2013

From Bruce Westbury, Aug 22 2013: (Start)

a(n) is also the number of nested pairs of Dyck paths of length n starting and ending at the origin;

a(n) is also the number of 3-noncrossing perfect matchings on 2n points;

a(n) is also the number of 2-triangulations on n-gon;

a(n) is also the dimension of the invariant subspace of 2n-th tensor power of the spin representation of Spin(5);

a(n) is also the dimension of the invariant subspace of 2n-th tensor power of the defining representation of Sp(4). (End)

a(-1) = -3/2, a(-2) = -1/4 makes some formulas true for all n in Z. - Michael Somos, Oct 02 2014

a(n) is the number of uniquely sorted permutations of length 2n+1 that avoid the pattern 312. - Colin Defant, Jun 08 2019

Examples

			Example: a(2)=3 counts EWEW, EEWW, ENSW.
G.f. = 1 + x + 3*x^2 + 14*x^3 + 84*x^4 + 594*x^5 + 4719*x^6 + 40898*x^7 + ...
		

References

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• 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.
• Jonathan Bloom and Sergi Elizalde, Pattern avoidance in matchings and partitions, arXiv preprint arXiv:1211.3442 [math.CO], 2012. See Table 1.
• N. Bonichon, A bijection between realizers of maximal plane graphs and pairs of non-crossing Dyck paths, Discr. Math., 298 (2005), 104-114.
• Matteo Cervetti and Luca Ferrari, Pattern avoidance in the matching pattern poset, arXiv:2009.01024 [math.CO], 2020. See Section 2.
• W. Y. C. Cheng, E. Y. P. Deng, R. R. X. Du, R. P. Stanley and C. H. Yan, Crossings and nestings of matchings and partitions, arXiv:math/0501230 [math.CO], 2005.
• S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 183).
• C. P. Davis-Stober, A bijection between a set of lexicographic semiorders and pairs of non-crossing Dyck paths, Journal of Mathematical Psychology, 54, 471-474.
• Myriam de Sainte-Catherine, Couplages et Pfaffiens en Combinatoire, Physique et Informatique, PhD Dissertation, Université Bordeaux I, 1983. (Annotated scanned copy of pages III.42-III.45)
• Colin Defant, Catalan Intervals and Uniquely Sorted Permutations, arXiv:1904.02627 [math.CO], 2019.
• Stefan Felsner, Eric Fusy, Marc Noy, and David Orden, Bijections for Baxter families and related objects, J. Combin. Theory Ser. A, 118(3):993-1020, 2011. See Theorem 8.4.
• E. Fusy, D. Poulalhon, and G. Schaeffer, Bijective counting of plane bipolar orientations and Schnyder words, Eur. J. Combinat. 30 (2009) 1646-1658, eq (2).
• Juan B. Gil, Peter R. W. McNamara, Jordan O. Tirrell, and Michael D. Weiner, From Dyck paths to standard Young tableaux, arXiv:1708.00513 [math.CO], 2017.
• D. Gouyou-Beauchamps, Chemins sous-diagonaux et tableaux de Young, pp. 112-125 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, Springer 1986.
• D. Gouyou-Beauchamps, Chemins sous-diagonaux et tableaux de Young, pp. 112-125 of "Combinatoire Enumerative (Montreal 1985)", Lect. Notes Math. 1234, Springer, 1986. (Annotated scanned copy)
• D. Gouyou-Beauchamps. Standard Young tableaux of height 4 and 5, European J. Combin. 10 (1989) 69 - 82.
• Nicholas M. Katz, A Note on Random Matrix Integrals, Moment Identities, and Catalan Numbers, preprint, 2015; Mathemtika, Volume 62, Issue 3 2016 , pp. 811-817.
• Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, arXiv:0803.4462 [math.NT], 2008-2010.
• Alec Mihailovs, Enumeration of walks on lattices, arXiv:math/9803128 (1998).
• Heinrich Niederhausen, A Note on the Enumeration of Diffusion Walks in the First Octant by Their Number of Contacts with the Diagonal, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.3.
• Maya Sankar, Further Bijections to Pattern-Avoiding Valid Hook Configurations, arXiv:1910.08895 [math.CO], 2019.
• Robert Scherer, Topics in Number Theory and Combinatorics, Ph. D. Dissertation, Univ. of California Davis (2021).
• Index entries for sequences related to Young tableaux.

Crossrefs

A column of the triangle in A179898. A diagonal of the triangle in A185249.

Row sums of A193691, A193692. - Alois P. Heinz, Aug 03 2011

See A138349 for another version.

Cf. A000108, A006149, A006150, A248152.

Programs

LiE

p_tensor(2*n,[0,1],B2)|[0,0]

LiE

p_tensor(2*n,[1,0],C2)|[0,0]

Magma

[6*Factorial(2*n)*Factorial(2*n+2)/(Factorial(n)*Factorial(n+1)* Factorial(n+2)*Factorial(n+3)): n in [0..25]]; // Vincenzo Librandi, Aug 04 2011

Mathematica

CoefficientList[ Series[ HypergeometricPFQ[ {1, 1/2, 3/2}, {3, 4}, 16 x], {x, 0, 19}], x]
a[ n_] := If[ n < 1, Boole[n == 0], Det[ Table[ Binomial[i + 1, j - i + 2], {i, n}, {j, n}]]]; (* Michael Somos, Feb 25 2014 *) (* slight modification of David Callan formula *)
a[ n_] := 12 * 4^n * (2*n-1)!! * (2*n+1)!! / ((n+2)! * (n+3)!); (* Michael Somos, Oct 02 2014 *)

PARI

a(n)=6*binomial(2*n+2,n)*(2*n)!/(n+1)!/(n+3)! \\ Charles R Greathouse IV, Aug 04 2011

PARI

{a(n) = if( n<0, if( n<-2, 0, [-3/2, -1/4][-n]), 6 * (2*n)! * (2*n+2)! / (n! * (n+1)! * (n+2)! * (n+3)!))}; /* Michael Somos, Oct 02 2014 */

Formula

G.f.: 3F2( [ 1, 1/2, 3/2 ]; [ 3, 4 ]; 16 x ).

a(n) = 6*(2*n)!*(2*n+2)!/(n!*(n+1)!*(n+2)!*(n+3)!) (Mihailovs).

a(n) = Det[Table[binomial[i+1, j-i+2], {i, 1, n}, {j, 1, n}]]. - David Callan, Jul 20 2005

a(n) = b(n)b(n+1)/6 where b(n) is the superballot number A007054. - David Callan, Feb 01 2007

a(n) = A000108(n)*A000108(n+2) - A000108(n+1)^2. - Philippe Deléham, Apr 11 2007

G.f.: (1 + 6*x - hypergeom([-1/2,-3/2],[2],16*x))/(4*x^2). - Mark van Hoeij, Nov 02 2009

From Michael Somos, Oct 02 2014: (Start)

a(n) = 12 * 4^n * (2*n-1)!! * (2*n+1)!! / ((n+2)! * (n+3)!).

D-finite with recurrence 0 = a(n) * 4*(2*n+1)*(2*n+3) - a(n+1) * (n+3)*(n+4) for all n in Z.

0 = a(n)*(+65536*a(n+2) - 72192*a(n+3) + 10296*a(n+4)) + a(n+1)*(-1536*a(n+2) - 1632*a(n+3) - 282*a(n+4)) + a(n+2)*(+40*a(n+2) - 6*a(n+3) + a(n+4)) for all n in Z.

0 = a(n)^2*a(n+2)*(+1792*a(n+1) - 882*a(n+2)) + a(n)*a(n+1)^2*(+768*a(n+1) + 580*a(n+2)) + 7*a(n)*a(n+1)*a(n+2)^2 +a(n+1)^3*(-18*a(n+1) + 3*a(n+2)) for all n in Z. (End)

a(n) ~ 3 * 2^(4*n+3) / (Pi * n^5). - Vaclav Kotesovec, Feb 10 2015

From Peter Bala, Feb 22 2023: (Start)

a(n) = (12*(2*n - 1)/((n + 1)(n + 2)(n + 3))) * Catalan(n-1)*Catalan(n+1) for n >= 1.

a(n) = Product_{1 <= i <= j <= n-1} (i + j + 4)/(i + j).

a(n) = (1/2^(n-1)) * Product_{1 <= i <= j <= n-1} (i + j + 4)/(i + j - 1) for n >= 1. (End)

Sum_{n>=0} a(n)/16^n = 88 - 4096/(15*Pi). - Amiram Eldar, May 06 2023

Extensions

More terms from James Sellers, Dec 24 1999

Corrected by Vladeta Jovovic, May 23 2004

Better definition from David Callan, Sep 18 2007

Definition simplified by N. J. A. Sloane, Nov 30 2020