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A001003 Schroeder's second problem (generalized parentheses); also called super-Catalan numbers or little Schroeder numbers.
(Formerly M2898 N1163)

%I M2898 N1163

%S 1,1,3,11,45,197,903,4279,20793,103049,518859,2646723,13648869,

%T 71039373,372693519,1968801519,10463578353,55909013009,300159426963,

%U 1618362158587,8759309660445,47574827600981,259215937709463,1416461675464871

%N Schroeder's second problem (generalized parentheses); also called super-Catalan numbers or little Schroeder numbers.

%C If you are looking for the Schroeder numbers (a.k.a. large Schroder numbers, or big Schroeder numbers), see A006318.

%C There are two schools of thought about the index for the first term. I prefer the indexing a(0) = a(1) = 1, a(2) = 3, a(3) = 11, etc.

%C a(n) = number of ways to insert parentheses in a string of n+1 symbols. The parentheses must be balanced but there is no restriction on the number of pairs of parentheses. The number of letters inside a pair of parentheses must be at least 2. Parentheses enclosing the whole string are ignored.

%C Also length of list produced by a variant of the Catalan producing iteration: replace each integer k with the list 0,1,..,k,k+1,k,...,1,0 and get the length a(n) of the resulting (flattened) list after n iterations. - _Wouter Meeussen_, Nov 11 2001

%C Stanley gives several other interpretations for these numbers.

%C Number of Schroeder paths of semilength n-1 (i.e., lattice paths from (0,0) to (2n-2,0), with steps H=(2,0), U=(1,1) and D(1,-1) and not going below the x-axis) with no peaks at level 1. Example: a(3)=3 because among the six Schroeder paths of semilength two HH, UHD, UUDD, HUD, UDH and UDUD, only the first three have no peaks at level 1. - _Emeric Deutsch_, Dec 27 2003

%C a(n+1) = number of Dyck n-paths in which the interior vertices of the ascents are colored white or black. - _David Callan_, Mar 14 2004

%C Number of possible schedules for n time slots in the first-come first-served (FCFS) printer policy.

%C Also row sums of A086810, A033282. - _Philippe Deléham_, May 09 2004

%C a(n+1) = number of pairs (u,v) of same-length compositions of n, 0's allowed in u but not in v and u dominates v (meaning u_1 >= v_1, u_1 + u_2 >= v_1 + v_2 and so on). For example, with n=2, a(3) counts (2,2), (1+1,1+1), (2+0,1+1). - _David Callan_, Jul 20 2005

%C The big Schroeder number (A006318) is the number of Schroeder paths from (0,0) to (n,n) (subdiagonal paths with steps (1,0) (0,1) and (1,1)). These paths fall in two classes: those with steps on the main diagonal and those without. These two classes are equinumerous and the number of paths in either class is the little Schroeder number a(n) (half the big Schroeder number). - Marcelo Aguiar (maguiar(AT)math.tamu.edu), Oct 14 2005

%C With offset 0, a(n) = number of (colored) Motzkin (n-1)-paths with each upstep U getting one of 2 colors and each flatstep F getting one of 3 colors. Example. With their colors immediately following upsteps/flatsteps, a(2) = 3 counts F1, F2, F3 and a(3)=11 counts U1D, U2D, F1F1, F1F2, F1F3, F2F1, F2F2, F2F3, F3F1, F3F2, F3F3. - _David Callan_, Aug 16 2006

%C Triangle A144156 has row sums = A006318 with left border A001003. - _Gary W. Adamson_, Sep 12 2008

%C Shifts left when INVERT transform applied twice. - _Alois P. Heinz_, Apr 01 2009

%C a(n) = A144944(n,n) = A186826(n,0). - _Reinhard Zumkeller_, May 11 2013

%C Number of increasing tableaux of shape (n,n). An increasing tableau is a semistandard tableaux with strictly increasing rows and columns, and set of entries an initial segment of the positive integers. Example: a(2) = 3 because of the three tableaux (12)(34), (13)(24), (12)(23). - _Oliver Pechenik_, Apr 22 2014

%C Number of ordered trees with no vertex of outdegree 1 and having n+1 leaves (called sometimes Schröder trees). - _Emeric Deutsch_, Dec 13 2014

%C Number of dissections of a convex (n+2)-gon by nonintersecting diagonals. Example: a(2)=3 because for a square ABCD we have (i) no diagonal, (ii) dissection with diagonal AC, and (iii) dissection with diagonal BD. - _Emeric Deutsch_, Dec 13 2014

%C The little Schroeder numbers are the moments of the Marchenko-Pastur law for the case c=2 (although the moment m0 is 1/2 instead of 1): 1/2, 1, 3, 11, 45, 197, 903, ... - _Jose-Javier Martinez_, Apr 07 2015

%C Number of generalized Motzkin paths with no level steps at height 0, from (0,0) to (2n,0), and consisting of steps U=(1,1), D=(1,-1) and H2=(2,0). For example, for n=3, we have the 11 paths: UDUDUD, UUDDUD, UDUUDD, UH2DUD, UDUH2D, UH2H2D, UUDUDD, UUUDDD, UUH2DD, UUDH2D, UH2UDD. - _José Luis Ramírez Ramírez_, Apr 20 2015

%C REVERT transform of A225883. - _Vladimir Reshetnikov_, Oct 25 2015

%D D. Arques and A. Giorgetti, Une bijection geometrique entre une famille d'hypercartes et une famille de polygones enumerees par la serie de Schroeder, Discrete Math., 217 (2000), 17-32.

%D P Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.

%D N. Bernasconi et al., On properties of random dissections and triangulations, Combinatorica, 30 (6) (2010), 627-654.

%D Cameron, Peter J. Some treelike objects. Quart. J. Math. Oxford Ser. (2) 38 (1987), no. 150, 155--183. MR0891613 (89a:05009). See p. 155, also p. 179, line -9. - _N. J. A. Sloane_, Apr 18 2014

%D P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102; also in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.

%D C. Coker, A family of eigensequences, Discrete Math. 282 (2004), 249-250.

%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 57.

%D D. E. Davenport, L. W. Shapiro and L. C. Woodson, The Double Riordan Group, The Electronic Journal of Combinatorics, 18(2) (2012), #P33. - From _N. J. A. Sloane_, May 11 2012

%D E. Deutsch, A bijective proof of an equation linking the Schroeder numbers, large and small, Discrete Math., 241 (2001), 235-240.

%D Doslic, Tomislav and Veljan, Darko. Logarithmic behavior of some combinatorial sequences. Discrete Math. 308 (2008), no. 11, 2182--2212. MR2404544 (2009j:05019) - From _N. J. A. Sloane_, May 01 2012

%D Drmota, Michael; de Mier, Anna; Noy, Marc. Extremal statistics on non-crossing configurations. Discrete Math. 327 (2014), 103--117. MR3192420. See Eq. (2). - _N. J. A. Sloane_, May 18 2014

%D I. M. H. Etherington, On non-associative combinations, Proc. Royal Soc. Edinburgh, 59 (Part 2, 1938-39), 153-162.

%D I. M. H. Etherington, Some problems of non-associative combinations (I), Edinburgh Math. Notes, 32 (1940), pp. i-vi. Part II is by A. Erdelyi and I. M. H. Etherington, and is on pages vii-xiv of the same issue.

%D P. Flajolet and M. Noy, Analytic combinatorics of non-crossing permutations, Discrete Math., 204 (1999), 203-229.

%D D. Foata and D. Zeilberger, A classic proof of a recurrence for a very classical sequence, J. Comb Thy A 80 380-384 1997

%D Wolfgang Gatterbauer, Dan Suciu, Dissociation and propagation for approximate lifted inference with standard relational database management systems, The VLDB Journal, February 2017, Volume 26, Issue 1, pp 5-30; DOI 10.1007/s00778-016-0434-5

%D Ivan Geffner, Marc Noy, Counting Outerplanar Maps, Electronic Journal of Combinatorics 24(2) (2017), #P2.3

%D D. Gouyou-Beauchamps and B. Vauquelin, Deux proprietes combinatoires des nombres de Schroeder, Theor. Inform. Appl., 22 (1988), 361-388.

%D N. S. S. Gu, N. Y. Li and T. Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.

%D P.-Y. Huang, S.-C. Liu, Y.-N. Yeh, Congruences of Finite Summations of the Coefficients in certain Generating Functions, The Electronic Journal of Combinatorics, 21 (2014), #P2.45.

%D M. Klazar, On numbers of Davenport-Schinzel sequences, Discr. Math., 185 (1998), 77-87.

%D D. E. Knuth, The Art of Computer Programming, Vol. 1, various sections (e.g. p. 534 of 2nd ed.).

%D D. E. Knuth, The Art of Computer Programming, Vol. 1, (p. 539 of 3rd ed.).

%D D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section, Problem 66, p. 479.

%D J. S. Lew, Polynomial enumeration of multidimensional lattices, Math. Systems Theory, 12 (1979), 253-270.

%D Ana Marco, J.-J. Martinez, A total positivity property of the Marchenko-Pastur Law, Electronic Journal of Linear Algebra, 30 (2015), #7.

%D T. S. Motzkin, Relations between hypersurface cross ratios and a combinatorial formula for partitions of a polygon, for permanent preponderance and for non-associative products, Bull. Amer. Math. Soc., 54 (1948), 352-360.

%D L. Ozsvart, Counting ordered graphs that avoid certain subgraphs, Discr. Math., 339 (2016), 1871-1877.

%D Feng Qi, BN Guo, Some explicit and recursive formulas of the large and little Schröder numbers, Arab Journal of Mathematical Sciences, June 2016; https://doi.org/10.1016/j.ajmsc.2016.06.002

%D J. Riordan, Combinatorial Identities, Wiley, 1968, p. 168.

%D E. Schroeder, Vier combinatorische Probleme, Zeit. f. Math. Phys., 15 (1870), 361-376.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see page 178; see page 239, Exercise 6.39b.

%D R. P. Stanley, Hipparcus, Plutarch, Schröder and Hough, Amer. Math. Monthly, 104, 1997, 344-350.

%D H. N. V. Temperley and D. G. Rogers, A note on Baxter's generalization of the Temperley-Lieb operators, pp. 324-328 of Combinatorial Mathematics (Canberra, 1977), Lect. Notes Math. 686, 1978.

%D I. Vardi, Computational Recreations in Mathematica. Addison-Wesley, Redwood City, CA, 1991, p. 198.

%D E. X. W. Xia and O. X. M. Yao, A Criterion for the Log-Convexity of Combinatorial Sequences, The Electronic Journal of Combinatorics, 20 (2013), #P3.

%H Seiichi Manyama, <a href="/A001003/b001003.txt">Table of n, a(n) for n = 0..1312</a> (first 200 terms from T. D. Noe)

%H J. Abate, W. Whitt, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Whitt/whitt2.html">Integer Sequences from Queueing Theory </a>, J. Int. Seq. 13 (2010), 10.5.5, p_n(1).

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H Marcelo Aguiar and Walter Moreira, <a href="http://arxiv.org/abs/math/0510169">Combinatorics of the free Baxter algebra</a>, arXiv:math/0510169 [math.CO], 2005-2007, see Corollary 3.3.iii.

%H A. Bacher, <a href="http://arxiv.org/abs/1301.1365">Directed and multi-directed animals on the square lattice with next nearest neighbor edges</a>, arXiv preprint arXiv:1301.1365 [math.CO], 2013. - From _N. J. A. Sloane_, Feb 04 2013

%H Axel Bacher, <a href="https://arxiv.org/abs/1802.06030">Improving the Florentine algorithms: recovering algorithms for Motzkin and Schröder paths</a>, arXiv:1802.06030 [cs.DS], 2018.

%H C. Banderier, M. Bousquet-Mélou, A. Denise, P. Flajolet, D. Gardy and D. Gouyou-Beauchamps, <a href="http://dx.doi.org/10.1016/S0012-365X(01)00250-3">Generating functions for generating trees</a>, Discrete Mathematics 246(1-3), March 2002, pp. 29-55.

%H C. Banderier and D. Merlini, <a href="http://www.dsi.unifi.it/~merlini/poster.ps">Lattice paths with an infinite set of jumps</a>, FPSAC02, Melbourne, 2002.

%H E. Barcucci, E. Pergola, R. Pinzani and S. Rinaldi, <a href="http://www.mat.univie.ac.at/~slc/wpapers/s46rinaldi.html">ECO method and hill-free generalized Motzkin paths</a>, Séminaire Lotharingien de Combinatoire, B46b (2001), 14 pp.

%H Paul Barry, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Barry/barry91.html">On Integer-Sequence-Based Constructions of Generalized Pascal Triangles</a>, J. Integer Sequ., Vol. 9 (2006), Article 06.2.4.

%H P. Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Barry1/barry95r.html">Generalized Catalan Numbers, Hankel Transforms and Somos-4 Sequences </a>, J. Int. Seq. 13 (2010) #10.7.2.

%H P. Barry, <a href="http://arxiv.org/abs/1311.7161">Comparing two matrices of generalized moments defined by continued fraction expansions</a>, arXiv preprint arXiv:1311.7161 [math.CO], 2013 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Barry3/barry291.html">J. Int. Seq. 17 (2014) # 14.5.1</a>.

%H Paul Barry, <a href="https://arxiv.org/abs/1802.03443">On a transformation of Riordan moment sequences</a>, arXiv:1802.03443 [math.CO], 2018.

%H Paul Barry and Aoife Hennessy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Barry2/barry190r.html">Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths</a>, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8.

%H Karin Baur, P. P. Martin, <a href="http://arxiv.org/abs/1601.05080">The fibres of the Scott map on polygon tilings are the flip equivalence classes</a>, arXiv preprint arXiv:1601.05080 [math.CO], 2016.

%H Karin Baur, Paul P. Martin, <a href="https://arxiv.org/abs/1711.04986">A generalised Euler-Poincaré formula for associahedra</a>, arXiv:1711.04986 [math.CO], 2017.

%H Arkady Berenstein, Vladimir Retakh, Christophe Reutenauer and Doron Zeilberger, <a href="http://arxiv.org/abs/1206.4225">The Reciprocal of Sum_{n >= 0} a^n b^n for non-commuting a and b, Catalan numbers and non-commutative quadratic equations</a>, arXiv preprint arXiv:1206.4225 [math.CO], 2012.

%H F. R. Bernhart & N. J. A. Sloane, <a href="/A006343/a006343.pdf">Emails, April-May 1994</a>

%H D. Birmajer, J. B. Gil, M. D. Weiner, <a href="http://arxiv.org/abs/1503.05242">Colored partitions of a convex polygon by noncrossing diagonals</a>, arXiv preprint arXiv:1503.05242 [math.CO], 2015.

%H J. Bloom, A, Burstein, <a href="http://arxiv.org/abs/1410.0230">Egge triples and unbalanced Wilf-equivalence</a>, arXiv preprint arXiv:1410.0230 [math.CO], 2014.

%H Miklós Bóna, Cheyne Homberger, Jay Pantone, and Vince Vatter, <a href="http://arxiv.org/abs/1310.7003">Pattern-avoiding involutions: exact and asymptotic enumeration</a>, arxiv:1310.7003 [math.CO], 2013.

%H H. Bottomley, <a href="/A001003/a001003.gif">Illustration of initial terms</a>

%H Mathilde Bouvel, Cedric Chauve, Marni Mishna and Dominique Rossin, <a href="http://arxiv.org/abs/1201.0940">Average-case analysis of perfect sorting by reversals</a>, arXiv preprint arXiv:1201.0940 [cs.DM], 2012.

%H Kevin Brown, <a href="http://www.mathpages.com/home/kmath397/kmath397.htm">Hipparchus on Compound Statements</a>, 1994-2010.

%H H. Cambazard, N. Catusse, <a href="http://arxiv.org/abs/1512.06649">Fixed-Parameter Algorithms for Rectilinear Steiner tree and Rectilinear Traveling Salesman Problem in the Plane</a>, arXiv preprint arXiv:1512.06649 [cs.DS], 2015.

%H Naiomi T. Cameron and Asamoah Nkwanta, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL8/Cameron/cameron46.html">On Some (Pseudo) Involutions in the Riordan Group</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.7.

%H F. Cazals, <a href="http://algo.inria.fr/libraries/autocomb/NCC-html/NCC.html">Combinatorics of Non-Crossing Configurations</a>, Studies in Automatic Combinatorics, Volume II (1997).

%H Grégory Chatel, Vincent Pilaud, Viviane Pons, <a href="https://arxiv.org/abs/1701.07995">The weak order on integer posets</a>, arXiv:1701.07995 [math.CO], 2017.

%H W. Y. C. Chen, T. Mansour and S. H. F. Yan, <a href="http://arXiv.org/abs/math.CO/0504342">Matchings avoiding partial patterns</a>, arXiv:math/0504342 [math.CO], 2005.

%H William Y. C. Chen and Carol J. Wang, <a href="http://arxiv.org/abs/1009.0176">Noncrossing Linked Partitions and Large (3, 2)-Motzkin Paths</a>, Discrete Math., 312 (2012), 1918-1922. - _N. J. A. Sloane_, Jun 08 2012

%H Z. Chen, H. Pan, <a href="http://arxiv.org/abs/1608.02448">Identities involving weighted Catalan-Schroder and Motzkin Paths</a>, arXiv:1608.02448 [math.CO] (2016), eq (1.13) a=1, b=2.

%H J. Cigler, <a href="http://arxiv.org/abs/1210.0372">Ramanujan's q-continued fractions and Schröder-like numbers</a>, arXiv:1210.0372 [math.HO], 2012. - _N. J. A. Sloane_, Dec 29 2012

%H J. Cigler, <a href="http://arxiv.org/abs/1211.0816">Hankel determinants of some polynomial sequences</a>, arXiv:1211.0816 [math.CO], 2012.

%H D. Drake, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Drake/drake.html">Bijections from Weighted Dyck Paths to Schröder Pathss</a, J. Int. Seq. 13 (2010) # 10.9.2

%H I. M. H. Etherington, <a href="http://www.jstor.org/stable/3605743">Non-associate powers and a functional equation</a>, Math. Gaz. 21 (1937), 36-39; addendum 21 (1937), 153.

%H I. M. H. Etherington, <a href="/A000108/a000108_14.pdf">Some problems of non-associative combinations (I)</a>, Edinburgh Math. Notes, 32 (1940), pp. i-vi. [Annotated scanned copy]. Part II [not scanned] is by A. Erdelyi and I. M. H. Etherington, and is on pages vii-xiv of the same issue.

%H S.-P. Eu and T.-S. Fu, <a href="http://arXiv.org/abs/math.CO/0412041">A simple proof of the Aztec diamond problem</a>, arXiv:math/0412041 [math.CO], 2004.

%H P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; see pages 69, 474-475, etc.

%H D. Foata and D. Zeilberger, <a href="http://arXiv.org/abs/math.CO/9805015">A Classic Proof of a Recurrence for a Very Classical Sequence</a>, arXiv:math/9805015 [math.CO], 1998.

%H D. Foata and D. Zeilberger, <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/classic.pdf">A classic proof of a recurrence for a very classical sequence</a>

%H N. Gabriel, K. Peske, L. Pudwell, S. Tay, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Pudwell/pudwell.html">Pattern Avoidance in Ternary Trees</a>, J. Int. Seq. 15 (2012) # 12.1.5

%H W. Gatterbauer, D. Suciu, <a href="http://arxiv.org/abs/1412.1069">Approximate Lifted Inference with Probabilistic Databases</a>, arXiv preprint arXiv:1412.1069 [cs.DB], 2014.

%H Evangelos Georgiadis, Akihiro Munemasa and Hajime Tanaka, <a href="http://arxiv.org/abs/1101.1579">A note on super Catalan numbers</a>, arXiv:1101.1579 [math.CO], 2011-2012.

%H Olivier Gérard, <a href="/A001003/a001003a.pdf">Illustration of initial terms (a)</a>

%H Olivier Gérard, <a href="/A001003/a001003b.pdf">Illustration of initial terms (b)</a>

%H Étienne Ghys, <a href="http://arxiv.org/abs/1612.06373">A Singular Mathematical Promenade</a>, arXiv:1612.06373, 2016.

%H Samuele Giraudo, <a href="http://arxiv.org/abs/1603.01394">Pluriassociative algebras II: The polydendriform operad and related operads</a>, arXiv:1603.01394 [math.CO], 2016.

%H Li Guo and Jun Pei, <a href="http://arxiv.org/abs/1401.7386">Averaging algebras, Schroeder numbers and rooted trees</a>, arXiv preprint arXiv:1401.7386 [math.RA], 2014.

%H Guo-Niu Han, <a href="http://www-irma.u-strasbg.fr/~guoniu/papers/p77puzzle.pdf">Enumeration of Standard Puzzles</a>

%H Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a> [Cached copy]

%H Aoife Hennessy, <a href="http://repository.wit.ie/1693/1/AoifeThesis.pdf">A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths</a>, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.

%H Cheyne Homberger, <a href="http://arxiv.org/abs/1410.2657">Patterns in Permutations and Involutions: A Structural and Enumerative Approach</a>, arXiv preprint 1410.2657 [math.CO], 2014.

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=42">Encyclopedia of Combinatorial Structures 42</a>

%H St. John Kettle, <a href="/A001003/a001003_5.pdf">Letter to N. J. A. Sloane</a>, 1982.

%H E. Krasko, A. Omelchenko, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i1p17">Brown's Theorem and its Application for Enumeration of Dissections and Planar Trees</a>, The Electronic Journal of Combinatorics, 22 (2015), #P1.17.

%H G. Kreweras, <a href="http://www.numdam.org/item?id=BURO_1973__20__3_0">Sur les hiérarchies de segments</a>, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973).

%H G. Kreweras, <a href="/A001844/a001844.pdf">Sur les hiérarchies de segments</a>, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)

%H G. Kreweras, <a href="http://archive.numdam.org/article/MSH_1976__53__5_0.pdf">Les préordres totaux compatibles avec un ordre partiel</a>, Math. Sci. Humaines No. 53 (1976), 5-30.

%H G. Kreweras, <a href="/A001003/a001003_1.pdf">Letter to N. J. A. Sloane</a>, 1978.

%H V. V. Kruchinin and D. V. Kruchinin, <a href="http://arxiv.org/abs/1206.0877">A Method for Obtaining Generating Function for Central Coefficients of Triangles</a>, arXiv preprint arXiv:1206.0877 [math.CO], 2012, and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Kruchinin/kruchinin5.html">J. Int. Seq. 15 (2012) #12.9.3</a>

%H V. Kurauskas, <a href="http://arxiv.org/abs/1504.08107">On graphs containing few disjoint excluded minors. Asymptotic number and structure of graphs containing few disjoint minors K_4</a>, arXiv preprint arXiv:1504.08107 [math.CO], 2015. See Section 8.1.

%H J. S. Lew, <a href="/A001003/a001003_4.pdf">Letter to N. J. A. Sloane, Sep 05 1978</a>

%H Huyile Liang, Jeffrey Remmel, Sainan Zheng, <a href="https://arxiv.org/abs/1710.05795">Stieltjes moment sequences of polynomials</a>, arXiv:1710.05795 [math.CO], 2017, see page 11.

%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/TheLostCatalanNumbers">The Lost Catalan Numbers And The Schröder Tableaux</a>.

%H A. Marco and J.-J. Martinez, <a href="http://repository.uwyo.edu/ela/vol30/iss1/7">A total positivity property of the Marchenko-Pastur Law</a>, Electronic Journal of Linear Algebra, Volume 30 (2015), pp. 106-117.

%H D. Merlini, R. Sprugnoli and M. C. Verri, <a href="http://www.dsi.unifi.it/~merlini/fun01.ps">Waiting patterns for a printer</a>, FUN with algorithm'01, Isola d'Elba, 2001.

%H W. Mlotkowski and K. A. Penson, <a href="http://arxiv.org/abs/1304.6544">The probability measure corresponding to 2-plane trees</a>, ArXiv preprint arXiv:1304.6544 [math.PR], 2013.

%H T. Motzkin, <a href="http://dx.doi.org/10.1090/S0002-9904-1945-08486-9">The hypersurface cross ratio</a>, Bull. Amer. Math. Soc., 51 (1945), 976-984.

%H T. S. Motzkin, <a href="http://dx.doi.org/10.1090/S0002-9904-1948-09002-4 ">Relations between hypersurface cross ratios and a combinatorial formula for partitions of a polygon, for permanent preponderance and for non-associative products</a>, Bull. Amer. Math. Soc., 54 (1948), 352-360.

%H G. Muntingh, <a href="http://arxiv.org/abs/1204.2709">Implicit Divided Differences, Little Schröder Numbers and Catalan Numbers</a>, arXiv preprint arXiv:1204.2709 [math.CO], 2012, and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Muntingh/muntingh2.html">J. Int. Seq. 15 (2012) #12.6.5</a>

%H Jean-Christophe Novelli and Jean-Yves Thibon, <a href="http://arxiv.org/abs/1209.5959">Duplicial algebras and Lagrange inversion</a>, arXiv preprint arXiv:1209.5959 [math.CO], 2012.

%H J.-C. Novelli, J.-Y. Thibon, <a href="http://arxiv.org/abs/1403.5962">Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions</a>, arXiv preprint arXiv:1403.5962 [math.CO], 2014.

%H P. Peart and W.-J. Woan, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/PEART/peart1.html">Generating Functions via Hankel and Stieltjes Matrices</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.1.

%H O. Pechenik, <a href="http://arxiv.org/abs/1209.1355">Cyclic sieving of increasing tableaux and small Schröder paths</a>, arXiv:1209.1355 [math.CO], 2012-2014.

%H O. Pechenik, <a href="http://dx.doi.org/10.1016/j.jcta.2014.04.002">Cyclic sieving of increasing tableaux and small Schröder paths</a>, J. Combin. Theory A, 125 (2014), 357-378.

%H R. A. Perez, <a href="http://www.ams.org/notices/201104/rtx110400558p.pdf">A brief but historic article of Siegel</a>, Notices AMS, 58 (2011), 558-566.

%H E. Pergola and R. A. Sulanke, <a href="https://cs.uwaterloo.ca/journals/JIS/PergolaSulanke/">Schroeder Triangles, Paths and Parallelogram Polyominoes</a>, J. Integer Sequences, 1 (1998), #98.1.7.

%H Vincent Pilaud, V. Pons, <a href="http://arxiv.org/abs/1606.09643">Permutrees</a>, arXiv preprint arXiv:1606.09643 [math.CO], 2016.

%H L. Pudwell, <a href="http://faculty.valpo.edu/lpudwell/slides/notredame.pdf">Pattern avoidance in trees (slides from a talk, mentions many sequences)</a>, 2012.

%H FENG QI, XIAO-TING SHI, AND BAI-NI GUO, <a href="https://www.researchgate.net/profile/Feng_Qi/publication/294259300">Integral representations of the large and little Schroder numbers</a>, Preprint, 2016.

%H E. Schröder, <a href="/A000108/a000108_9.pdf">Vier combinatorische Probleme</a>, Z. f. Math. Phys., 15 (1870), 361-376. [Annotated scanned copy]

%H L. W. Shapiro & N. J. A. Sloane, <a href="/A006318/a006318_1.pdf">Correspondence, 1976</a>

%H M. Shattuck, <a href="http://ami.ektf.hu/uploads/papers/finalpdf/AMI_42_from93to101.pdf">On the zeros of some polynomials with combinatorial coefficients</a>, Annales Mathematicae et Informaticae, 42 (2013) pp. 93-101.

%H N. J. A. Sloane, <a href="http://arXiv.org/abs/math.CO/0312448">The On-Line Encyclopedia of Integer Sequences</a>, arXiv:math/0312448 [math.CO], 2003.

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%H N. J. A. Sloane, <a href="/A001003/a001003_2.pdf">Illustrations of rooted planar trees with 2, 3, and 4 endpoints, illustrating a(1)=1, a(2)=3, a(4)=11.</a>

%H L. M. Smiley, <a href="http://arXiv.org/abs/math.CO/9907057">Variants of Schroeder Dissections</a>, arXiv:math/9907057 [math.CO], 1999.

%H Michael Z. Spivey and Laura L. Steil, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Spivey/spivey7.html">The k-Binomial Transforms and the Hankel Transform</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.

%H R. P. Stanley, <a href="http://www-math.mit.edu/~rstan/papers.html">Hipparchus, Plutarch, Schröder and Hough</a>, Am. Math. Monthly, Vol. 104, No. 4, p. 344, 1997.

%H Y. Sun, Z. Wang, <a href="http://dx.doi.org/10.1007/s00373-010-0950-9">Consecutive pattern avoidances in non-crossing trees</a>, Graph. Combinat. 26 (2010) 815-832

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Bracketing.html">Bracketing</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SuperCatalanNumber.html">Super Catalan Number</a>

%H Wen-jin Woan, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL8/Woan/woan11.html">A Recursive Relation for Weighted Motzkin Sequences</a> Journal of Integer Sequences, Vol. 8 (2005), Article 05.1.6.

%H Wen-jin Woan, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Woan/woan206.html">A Relation Between Restricted and Unrestricted Weighted Motzkin Paths</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.7.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Schr%C3%B6der%E2%80%93Hipparchus_number">Schroeder-Hipparchus numbers</a>.

%H <a href="/index/Par#parens">Index entries for sequences related to parenthesizing</a>

%H <a href="/index/Cor#core">Index entries for "core" sequences</a>

%F (n+1) * a(n) = (6*n-3) * a(n-1) - (n-2) * a(n-2) if n>1. a(0) = a(1) = 1.

%F a(n) = 3*a(n-1) + 2*A065096(n-2) (n>2). If g(x) = 1 + 3x + 11x^2 + 45x^3 + ... + a(n)*x^n + ..., then g(x) = 1 + 3(x*g(x)) + 2(x*g(x))^2, g(x)^2 = 1 + 6x + 31x^2 + 156x^3 + ... + A065096(n)*x^n + ... - _Paul D. Hanna_, Jun 10 2002

%F a(n+1) = -a(n) + 2*Sum_{k=1..n} a(k)*a(n+1-k). - _Philippe Deléham_, Jan 27 2004

%F a(n-2) = (1/(n-1))*Sum_{k=0..n-3} binomial(n-1,k+1)*binomial(n-3,k)*2^(n-3-k) for n >= 3 [G. Polya, Elemente de Math., 12 (1957), p. 115.] - _N. J. A. Sloane_, Jun 13 2015

%F G.f.: (1 + x - sqrt(1 - 6*x + x^2) )/(4*x) = 2/(1 + x + sqrt(1 - 6*x + x^2)).

%F a(n) ~ W*(3+sqrt(8))^n*n^(-3/2) where W = (1/4)*sqrt((sqrt(18)-4)/Pi) [See Knuth I, p. 534, or Perez. Note that the formula on line 3, page 475 of Flajolet and Sedgewick seems to be wrong - it has to be multiplied by 2^(1/4).] - _N. J. A. Sloane_, Apr 10 2011

%F The correct asymptotic for A001003 is a(n) ~ W*(3+sqrt(8))^n*n^(-3/2), where W = (1+sqrt(2))/(2*2^(3/4)*sqrt(Pi)) = 0.404947065905750651736243... Result in book by D. Knuth (p. 539 of 3rd edition, exercise 12) is for sequence b(n), but a(n) = b(n+1)/2. Therefore is asymptotic a(n) ~ b(n) * (3+sqrt(8))/2. - _Vaclav Kotesovec_, Sep 09 2012

%F The Hankel transform of this sequence gives A006125 = 1, 1, 2, 8, 64, 1024, ...; example: det([1, 1, 3, 11; 1, 3, 11, 45; 3, 11, 45, 197; 11, 45, 197, 903]) = 2^6 = 64. - _Philippe Deléham_, Mar 02 2004

%F a(n+1) = Sum_{k=0..floor((n-1)/2)} 2^k * 3^(n-1-2k) * binomial(n-1, 2k) * Catalan(k). This formula counts colored Dyck paths by the same parameter by which Touchard's identity counts ordinary Dyck paths: number of DDUs (U=up step, D=down step). See also Gouyou-Beauchamps reference. - _David Callan_, Mar 14 2004

%F a(n) = (1/(n+1))*Sum_{k=0..n} C(n+1, k)*C(2n-k, n)*(-1)^k*2^(n-k) [with offset 0]; a(n) = (1/(n+1))*Sum_{k=0..n} C(n+1, k+1)*C(n+k, k)*(-1)^(n-k)*2^k [with offset 0]; a(n) = Sum_{k=0..n} (1/(k+1))*C(n, k)*C(n+k, k)*(-1)^(n-k)*2^k [with offset 0]; a(n) = Sum_{k=0..n} A088617(n, k)*(-1)^(n-k)*2^k [with offset 0]. - _Paul Barry_, May 24 2005

%F E.g.f. of a(n+1) is exp(3*x)*BesselI(1, 2*sqrt(2)*x)/(sqrt(2)*x). - _Vladeta Jovovic_, Mar 31 2004

%F Reversion of (x-2*x^2)/(1-x) is g.f. offset 1.

%F For n>=1, a(n) = Sum_{k=0..n} 2^k*N(n, k) where N(n, k) = (1/n)*C(n, k)*C(n, k+1) are the Narayana numbers (A001263). - _Benoit Cloitre_, May 10 2003 [This formula counts colored Dyck paths by number of peaks, which is easy to see because the Narayana numbers count Dyck paths by number of peaks and the number of peaks determines the number of interior ascent vertices.]

%F a(n) = Sum_{k=0..n} A088617(n, k)*2^k*(-1)^(n-k). - _Philippe Deléham_, Jan 21 2004

%F For n > 0, a(n) = (1/(n+1)) * Sum_{k = 0 .. n-1} binomial(2*n-k, n)*binomial(n-1, k). This formula counts colored Dyck paths (as above) by number of white vertices. - _David Callan_, Mar 14 2004

%F a(n-1) = (d^(n-1)/dx^(n-1))((1-x)/(1-2*x))^n/n!|_{x=0}. (For a proof see the comment on the unsigned row sums of triangle A111785.)

%F a(n) = (1/n)*Sum_{k=1..n} binomial(n, k)*binomial(n+k, k-1) = hypergeom([1-n, n+2], [2], -1), n>=1. - _Wolfdieter Lang_, Sep 12 2005

%F a(n) = hypergeom([1-n, -n], [2], 2) for n>=0. - _Peter Luschny_, Sep 22 2014

%F a(m+n+1) = Sum_{k>=0} A110440(m, k)*A110440(n, k)*2^k = A110440(m+n, 0). - _Philippe Deléham_, Sep 14 2005

%F Sum over partitions formula (reference Schroeder paper p. 362, eq. (1) II). Number the partitions of n according to Abramowitz-Stegun pp. 831-832 (see reference under A105805) with k=1..p(n)= A000041(n). For n>=1: a(n-1) = Sum_{k=2..p(n)} A048996(n,k)*a(1)^e(k, 1)*a(1)^e(k, 2)*...*a(n-2)^e(k, n-1) if the k-th partition of n in the mentioned order is written as (1^e(k, 1), 2^e(k, 2), ..., (n-1)e(k, n-1)). Note that the first (k=1) partition (n^1) has to be omitted. - _Wolfdieter Lang_, Aug 23 2005

%F Starting (1, 3, 11, 45, ...), = row sums of triangle A126216 = A001263 * [1, 2, 4, 8, 16, ...]. - _Gary W. Adamson_, Nov 30 2007

%F From _Paul Barry_, May 15 2009: (Start)

%F G.f.: 1/(1+x-2x/(1+x-2x/(1+x-2x/(1+x-2x/(1-.... (continued fraction).

%F G.f.: 1/(1-x/(1-x-x/(1-x-x/(1-x-x/(1-... (continued fraction).

%F G.f.: 1/(1-x-2x^2/(1-3x-2x^2/(1-3x-2x^2/(1-... (continued fraction). (End)

%F G.f.: 1 / (1 - x / (1 - 2*x / (1 - x / (1 - 2*x / ... )))). - _Michael Somos_, May 19 2013

%F a(n) = (LegendreP(n+1,3)-3*LegendreP(n,3))/(4*n) for n>0. - _Mark van Hoeij_, Jul 12 2010 [This formula is mentioned in S.-J. Kettle's 1982 letter - see link. _N. J. A. Sloane_, Jun 13 2015]

%F From _Gary W. Adamson_, Jul 08 2011: (Start)

%F a(n) = upper left term in M^n, where M is the production matrix:

%F 1, 1, 0, 0, 0, 0, ...

%F 2, 2, 2, 0, 0, 0, ...

%F 1, 1, 1, 1, 0, 0, ...

%F 2, 2, 2, 2, 2, 0, ...

%F 1, 1, 1, 1, 1, 1, ...

%F ... (End)

%F From _Gary W. Adamson_, Aug 23 2011: (Start)

%F a(n) is the sum of top row terms of Q^(n-1), where Q is the infinite square production matrix:

%F 1, 2, 0, 0, 0, ...

%F 1, 1, 2, 0, 0, ...

%F 1, 1, 1, 2, 0, ...

%F 1, 1, 1, 1, 2, ...

%F ... (End)

%F Let h(t) = (1-t)^2/(2*(1-t)^2-1) = 1/(1-(2*t+3*t^2+4*t^3+...)), an o.g.f. for A003480, then for A001003 a(n) = (1/n!)*((h(t)*d/dt)^n) t, evaluated at t=0, with initial n=1. (Cf. A086810.) - _Tom Copeland_, Sep 06 2011

%F A006318(n) = 2*a(n) if n>0. - _Michael Somos_, Mar 31 2007

%F BINOMIAL transform is A118376 with offset 0. REVERT transform is A153881. INVERT transform is A006318. INVERT transform of A114710. HANKEL transform is A139685. PSUM transform is A104858. - _Michael Somos_, May 19 2013

%F G.f.: 1 + x/(Q(0) - x) where Q(k) = 1 + k*(1-x) - x - x*(k+1)*(k+2)/Q(k+1) ; (continued fraction). - _Sergei N. Gladkovskii_, Mar 14 2013

%F a(n)=(-1)^n*LegendreP(n,-1,-3)/sqrt(2), n > 0, LegendreP(n,a,b) is the Legendre function. - _Karol A. Penson_, Jul 06 2013

%F Integral representation as n-th moment of a positive weight function W(x) = W_a(x) + W_c(x), where W_a(x) = Dirac(x)/2, is the discrete (atomic) part, and W_c(x) = sqrt(8-(x-3)^2)/(4*Pi*x) is the continuous part of W(x) defined on (3 sqrt(8),3+sqrt(8)): a(n) = int( x^n*W_a(x), x=-eps..eps ) + int( x^n*W_c(x), x = 3-sqrt(8)..3+sqrt(8) ), for any eps>0, n>=0. W_c(x) is unimodal, of bounded variation and W_c(3-sqrt(8)) = W_c(3+sqrt(8)) = 0. Note that the position of the Dirac peak (x=0) lies outside support of W_c(x). - _Karol A. Penson_ and Wojciech Mlotkowski, Aug 05 2013

%e G.f. = 1 + x + 3*x^2 + 11*x^3 + 45*x^4 + 197*x^5 + 903*x^6 + 4279*x^7 + ...

%e a(2) = 3: abc, a(bc), (ab)c; a(3) = 11: abcd, (ab)cd, a(bc)d, ab(cd), (ab)(cd), a(bcd), a(b(cd)), a((bc)d), (abc)d, (a(bc))d, ((ab)c)d.

%e Sum over partitions formula: a(3) = 2*a(0)*a(2) + 1*a(1)^2 + 3*(a(0)^2)*a(1) + 1*a(0)^4 = 6 + 1 + 3 + 1 = 11.

%e a(4) = 45 since the top row of Q^3 = (11, 14, 12, 8, 0, 0, 0,...); (11 + 14 + 12 + 8) = 45.

%p t1 := (1/(4*x))*(1+x-sqrt(1-6*x+x^2)); series(t1,x,40);

%p invtr:= proc(p) local b; b:= proc(n) option remember; local i; `if`(n<1, 1, add(b(n-i) *p(i-1), i=1..n+1)) end end: a:='a': f:= (invtr@@2)(a): a:= proc(n) if n<0 then 1 else f(n-1) fi end: seq(a(n), n=0..30); # _Alois P. Heinz_, Apr 01 2009

%p # Computes n -> [a[0],a[1],..,a[n]]

%p A001003_list := proc(n) local j,a,w; a := array(0..n); a[0] := 1;

%p for w from 1 to n do a[w] := a[w-1]+2*add(a[j]*a[w-j-1],j=1..w-1) od;

%p convert(a,list) end: A001003_list(100); # _Peter Luschny_, May 17 2011

%t Table[Length[Flatten[Nest[ #/.a_Integer:> Join[Range[0, a + 1], Range[a, 0, -1]] &, {0}, n]]], {n, 0, 10}]; Sch[ 0 ] = Sch[ 1 ] = 1; Sch[ n_Integer ] := Sch[ n ] = (3(2n - 1)Sch[ n - 1 ] - (n - 2)Sch[ n - 2 ])/(n + 1); Array[ Sch, 24, 0]

%t a[n_] := Hypergeometric2F1[-n + 1, n + 2, 2, -1]; a[0] = 1; Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Nov 09 2011, after _Vladeta Jovovic_ *)

%t a[ n_] := SeriesCoefficient[ (1 + x - Sqrt[1 - 6 x + x^2]) / (4 x), {x, 0, n}]; (* _Michael Somos_, Aug 26 2015 *)

%t Table[(KroneckerDelta[n] - GegenbauerC[n+1, -1/2, 3])/4, {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 25 2015 *)

%o (PARI) {a(n) = if( n<1, n==0, sum( k=0, n, 2^k * binomial(n, k) * binomial(n, k-1) ) / (2*n))}; /* _Michael Somos_, Mar 31 2007 */

%o (PARI) {a(n) = my(A); if( n<1, n==0, n--; A = x * O(x^n); n! * simplify( polcoeff( exp(3*x + A) * besseli(1, 2*x * quadgen(8) + A), n)))}; /* _Michael Somos_, Mar 31 2007 */

%o (PARI) {a(n) = if( n<0, 0, n++; polcoeff( serreverse( (x - 2*x^2) / (1 - x) + x * O(x^n)), n))}; /* _Michael Somos_, Mar 31 2007 */

%o (Python) # The objective of this implementation is efficiency.

%o # n -> [a(0), a(1), ..., a(n)]

%o def A001003_list(n):

%o ....a = [0 for i in range(0, n)]

%o ....a[0] = 1

%o ....for w in range(1, n):

%o ........s = 0

%o ........for j in range(1, w):

%o ............s += a[j]*a[w-j-1]

%o ........a[w] = a[w-1]+2*s

%o ....return a

%o # _Peter Luschny_, May 17 2011

%o (Sage) # Generalized algorithm of L. Seidel

%o def A001003_list(n) :

%o D = [0]*(n+1); D[1] = 1/2

%o b = True; h = 2; R = [1]

%o for i in range(2*n-2) :

%o if b :

%o for k in range(h,0,-1) : D[k] += D[k-1]

%o h += 1;

%o else :

%o for k in range(1,h, 1) : D[k] += D[k-1]

%o R.append(D[h-1]);

%o b = not b

%o return R

%o A001003_list(24) # _Peter Luschny_, Jun 02 2012

%o (Haskell)

%o a001003 = last . a144944_row -- _Reinhard Zumkeller_, May 11 2013

%o (Python)

%o from gmpy2 import divexact

%o A001003 = [1, 1]

%o for n in range(3,10**3):

%o ....A001003.append(divexact(A001003[-1]*(6*n-9)-(n-3)*A001003[-2],n))

%o # _Chai Wah Wu_, Sep 01 2014

%Y See A000108, A001190, A001699, A000081 for other ways to count parentheses.

%Y Cf. A000311, A006318 (Schroeder numbers), A010683, A065096.

%Y Row sums of A033877.

%Y Cf. A054726, A059435, A025240, A080243, A085403, A086456, A086616, A035011.

%Y Right-hand column 1 of triangle A011117.

%Y See also convolution triangle A011117.

%Y Cf. A144156.

%K nonn,easy,nice,core

%O 0,3

%A _N. J. A. Sloane_

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