This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A009766 #473 Aug 11 2025 10:48:23
%S A009766 1,1,1,1,2,2,1,3,5,5,1,4,9,14,14,1,5,14,28,42,42,1,6,20,48,90,132,132,
%T A009766 1,7,27,75,165,297,429,429,1,8,35,110,275,572,1001,1430,1430,1,9,44,
%U A009766 154,429,1001,2002,3432,4862,4862,1,10,54,208,637,1638,3640,7072,11934
%N A009766 Catalan's triangle T(n,k) (read by rows): each term is the sum of the entries above and to the left, i.e., T(n,k) = Sum_{j=0..k} T(n-1,j).
%C A009766 The entries in this triangle (in its many forms) are often called ballot numbers.
%C A009766 T(n,k) = number of standard tableaux of shape (n,k) (n > 0, 0 <= k <= n). Example: T(3,1) = 3 because we have 134/2, 124/3 and 123/4. - _Emeric Deutsch_, May 18 2004
%C A009766 T(n,k) is the number of full binary trees with n+1 internal nodes and jump-length k. In the preorder traversal of a full binary tree, any transition from a node at a deeper level to a node on a strictly higher level is called a jump; the positive difference of the levels is called the jump distance; the sum of the jump distances in a given ordered tree is called the jump-length. - _Emeric Deutsch_, Jan 18 2007
%C A009766 The k-th diagonal from the right (k=1, 2, ...) gives the sequence obtained by asking in how many ways can we toss a fair coin until we first get k more heads than tails. The k-th diagonal has formula k(2m+k-1)!/(m!(m+k)!) and g.f. (C(x))^k where C(x) is the generating function for the Catalan numbers, (1-sqrt(1-4*x))/(2*x). - _Anthony C Robin_, Jul 12 2007
%C A009766 T(n,k) is also the number of order-decreasing and order-preserving full transformations (of an n-element chain) of waist k (waist (alpha) = max(Im(alpha))). - _Abdullahi Umar_, Aug 25 2008
%C A009766 Formatted as an upper right triangle (see tables) a(c,r) is the number of different triangulated planar polygons with c+2 vertices, with triangle degree c-r+1 for the same vertex X (c=column number, r=row number, with c >= r >= 1). - _Patrick Labarque_, Jul 28 2010
%C A009766 The triangle sums, see A180662 for their definitions, link Catalan's triangle, its mirror is A033184, with several sequences, see crossrefs. - _Johannes W. Meijer_, Sep 22 2010
%C A009766 The m-th row of Catalan's triangle consists of the unique nonnegative differences of the form binomial(m+k,m)-binomial(m+k,m+1) with 0 <= k <= m (See Links). - _R. J. Cano_, Jul 22 2014
%C A009766 T(n,k) is also the number of nondecreasing parking functions of length n+1 whose maximum element is k+1. For example T(3,2) = 5 because we have (1,1,1,3), (1,1,2,3), (1,2,2,3), (1,1,3,3), (1,2,3,3). - _Ran Pan_, Nov 16 2015
%C A009766 T(n,k) is the number of Dyck paths from (0,0) to (n+2,n+2) which start with n-k+2 east steps and touch the diagonal y=x only on the last north step. - _Felipe Rueda_, Sep 18 2019
%C A009766 T(n-1,k) for k < n is number of well-formed strings of n parenthesis pairs with prefix of exactly n-k opening parenthesis; T(n,n) = T(n,n-1). - _Hermann Stamm-Wilbrandt_, May 02 2021
%D A009766 William Feller, Introduction to Probability Theory and its Applications, vol. I, ed. 2, chap. 3, sect. 1,2.
%D A009766 Ki Hang Kim, Douglas G. Rogers, and Fred W. Roush, Similarity relations and semiorders. Proceedings of the Tenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1979), pp. 577-594, Congress. Numer., XXIII-XXIV, Utilitas Math., Winnipeg, Man., 1979. MR0561081 (81i:05013).
%D A009766 D. E. Knuth, TAOCP, Vol. 4, Section 7.2.1.6, Eq. 22, p. 451.
%D A009766 C. Krishnamachary and M. Bheemasena Rao, Determinants whose elements are Eulerian, prepared Bernoullian and other numbers, J. Indian Math. Soc., 14 (1922), 55-62, 122-138 and 143-146.
%D A009766 M. Bellon, Query 5467, L'Intermédiaire des Mathématiciens, 4 (1925), 11; H. Ory, 4 (1925), 120. - _N. J. A. Sloane_, Mar 09 2022
%D A009766 Andrzej Proskurowski and Ekaputra Laiman, Fast enumeration, ranking, and unranking of binary trees. Proceedings of the thirteenth Southeastern conference on combinatorics, graph theory and computing (Boca Raton, Fla., 1982). Congr. Numer. 35 (1982), 401-413.MR0725898 (85a:68152).
%D A009766 M. Welsch, Note #371, L'Intermédiaire des Mathématiciens, 2 (1895), pp. 235-237. - _N. J. A. Sloane_, Mar 02 2022
%H A009766 T. D. Noe, <a href="/A009766/b009766.txt">Rows n = 0..100 of triangle, flattened</a>
%H A009766 Erik Aas, Arvind Ayyer, Svante Linusson and Samu Potka, <a href="https://arxiv.org/abs/1902.02019">The exact phase diagram for a semipermeable TASEP with nonlocal boundary jumps</a>, arXiv:1902.02019 [cond-mat.stat-mech], 2019.
%H A009766 Ron M. Adin, E. Bagno, and Y. Roichman, <a href="https://arxiv.org/abs/1611.06979">Block decomposition of permutations and Schur-positivity</a>, arXiv:1611.06979 [math.CO], 2016-2017.
%H A009766 Kassie Archer, Abigail Bishop, Alexander Diaz-Lopez, Luis David Garcia Puente, Darren Glass, and Joel Louwsma, <a href="https://arxiv.org/abs/1903.01393">Arithmetical structures on bidents</a>, arXiv:1903.01393 [math.CO], 2019.
%H A009766 J. L. Arregui, <a href="https://arxiv.org/abs/math/0109108">Tangent and Bernoulli numbers</a> related to Motzkin and Catalan numbers by means of numerical triangles, arXiv:math/0109108 [math.NT], 2001.
%H A009766 Jean-Christophe Aval, <a href="http://arxiv.org/abs/0711.0906">Multivariate Fuss-Catalan numbers</a>, arXiv:0711.0906 [math.CO], 2007.
%H A009766 Jean-Christophe Aval, <a href="http://dx.doi.org/10.1016/j.disc.2007.08.100">Multivariate Fuss-Catalan numbers</a>, Discrete Math., 308 (2008), 4660-4669.
%H A009766 Axel Bacher, Antonio Bernini, Luca Ferrari, Benjamin Gunby, Renzo Pinzani, and Julian West, <a href="http://dx.doi.org/10.1016/j.disc.2013.12.011">The Dyck pattern poset</a> Discrete Math. 321 (2014), 12--23. MR3154009.
%H A009766 D. F. Bailey, <a href="http://www.maa.org/sites/default/files/D11233._F._Bailey.pdf">Counting arrangements of 1's and-1's</a>, Mathematics Magazine, 69 (1996): 128-131. See table on p. 129.
%H A009766 Elena Barcucci, Alberto Del Lungo, Elisa Pergola, and Renzo Pinzani, <a href="http://dx.doi.org/10.1016/S0012-365X(97)00122-2">A methodology for plane tree enumeration</a>, Proceedings of the 7th Conference on Formal Power Series and Algebraic Combinatorics (Noisy-le-Grand, 1995). Discrete Math. 180 (1998), no. 1-3, 45--64. MR1603693 (98m:05090).
%H A009766 E. Barcucci and M. C. Verri, <a href="http://dx.doi.org/10.1016/0012-365X(92)90117-X">Some more properties of Catalan numbers</a>, Discrete Math., 102 (1992), 229-237.
%H A009766 J.-L. Baril, C. Khalil, and V. Vajnovszki, <a href="https://arxiv.org/abs/2004.01812">Catalan and Schröder permutations sortable by two restricted stacks</a>, arXiv:2004.01812 [cs.DM], 2020.
%H A009766 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Barry1/barry97r2.html">Riordan Arrays, Orthogonal Polynomials as Moments, and Hankel Transforms</a>, J. Int. Seq. 14 (2011) # 11.2.2, example 3.
%H A009766 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Barry3/barry252.html">On the Inverses of a Family of Pascal-Like Matrices Defined by Riordan Arrays</a>, Journal of Integer Sequences, 16 (2013), #13.5.6.
%H A009766 Paul Barry, <a href="https://www.emis.de/journals/JIS/VOL22/Barry3/barry422.html">Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles</a>, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.
%H A009766 Paul Barry, <a href="https://arxiv.org/abs/2001.08799">Characterizations of the Borel triangle and Borel polynomials</a>, arXiv:2001.08799 [math.CO], 2020.
%H A009766 Paul Barry, <a href="https://arxiv.org/abs/2101.06713">On the inversion of Riordan arrays</a>, arXiv:2101.06713 [math.CO], 2021.
%H A009766 Paul Barry and A. Hennessy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Barry5/barry96s.html">Meixner-Type Results for Riordan Arrays and Associated Integer Sequences</a>, J. Int. Seq. 13 (2010) # 10.9.4, example 3.
%H A009766 F. R. Bernhart, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00054-0">Catalan, Motzkin and Riordan numbers</a>, Discr. Math., 204 (1999) 73-112.
%H A009766 A. Bernini, L. Ferrari, R. Pinzani, and J. West, <a href="http://arxiv.org/abs/1303.3785">The Dyck pattern poset</a>, arXiv:1303.3785 [math.CO], 2013.
%H A009766 N. Borie, <a href="http://arxiv.org/abs/1311.6292">Combinatorics of simple marked mesh patterns in 132-avoiding permutations</a>, arXiv:1311.6292 [math.CO], 2013.
%H A009766 M. Bousquet-Mélou and M. Petkovsek, <a href="https://doi.org/10.1016/S0012-365X(00)00147-3">Linear recurrences with constant coefficients: the multivariate case</a>, Discrete Math. 225 (2000), 51-75.
%H A009766 Jasper Bown, Peter Kagey, Alan Kappler, Michael E. Orrison, and Jayden Thadani, <a href="https://arxiv.org/abs/2507.11701">Preference-restricted parking functions</a>, arXiv:2507.11701 [math.CO], 2025. See p. 11.
%H A009766 Benjamin Braun, Hugo Corrales, Scott Corry, Luis David García Puente, Darren Glass, Nathan Kaplan, Jeremy L. Martin, Gregg Musiker, and Carlos E. Valencia, <a href="https://arxiv.org/abs/1701.06377">Counting Arithmetical Structures on Paths and Cycles</a>. arXiv:1701.06377 [math.CO], 2017.
%H A009766 E. H. M. Brietzke, <a href="http://dx.doi.org/10.1016/j.disc.2007.08.050">An identity of Andrews and a new method for the Riordan array proof of combinatorial identities</a>, Discrete Math., 308 (2008), 4246-4262.
%H A009766 S. Brlek, E. Duchi, E. Pergola, and S. Rinaldi, <a href="http://dx.doi.org/10.1016/j.disc.2004.07.019">On the equivalence problem for succession rules</a>, Discr. Math., 298 (2005), 142-154.
%H A009766 Hong Duc Bui, <a href="https://arxiv.org/abs/2504.09615">Counting Number of Triangulations of Point Sets: Reinterpreting and Generalizing the Triangulation Polynomials</a>, arXiv:2504.09615 [cs.CG], 2025. See p. 4.
%H A009766 Steve Butler, R. Graham, and C. H. Yan, <a href="http://www.math.ucsd.edu/~ronspubs/17_03_parking.pdf">Parking distributions on trees</a>, European Journal of Combinatorics 65 (2017), 168-185.
%H A009766 R. J. Cano, <a href="http://oeis.org/w/images/b/bc/CatalanBooks.pdf">Catalan's books</a>
%H A009766 L. Carlitz, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/10-5/carlitz7.pdf">Sequences, paths, ballot numbers</a>
%H A009766 Douglas M. Chen, <a href="https://arxiv.org/abs/2311.15699">On the Structure of Permutation Invariant Parking</a>, arXiv:2311.15699 [math.CO], 2023. See p. 16.
%H A009766 Lapo Cioni and Luca Ferrari, <a href="https://arxiv.org/abs/2102.07628">Preimages under the Queuesort algorithm</a>, arXiv preprint arXiv:2102.07628 [math.CO], 2021; Discrete Math., 344 (2021), #112561.
%H A009766 Ari Cruz, Pamela E. Harris, Kimberly J. Harry, Jan Kretschmann, Matt McClinton, Alex Moon, John O. Museus, and Eric Redmon, <a href="https://arxiv.org/abs/2312.16786">On some discrete statistics of parking functions</a>, arXiv:2312.16786 [math.CO], 2023.
%H A009766 Italo J. Dejter, <a href="http://home.coqui.net/dejterij/aneliese.pdf">A new approach to the middle levels via a Catalan-number system of numeration</a>, 2015.
%H A009766 Italo J. Dejter, <a href="http://home.coqui.net/dejterij/anumeral.pdf">A numeral system for the middle levels</a>, preprint, 2014. [See Section 2. - _N. J. A. Sloane_, Apr 06 2014]
%H A009766 Italo J. Dejter, <a href="http://home.coqui.net/dejterij/acson.pdf">Dihedral-symmetry middle-levels problem via a Catalan system of numeration</a>, preprint, 2015.
%H A009766 Italo J. Dejter, <a href="https://www.researchgate.net/publication/245576352_On_a_lexical_tree_for_the_middle-levels_graph_problem">The role of restricted growth strings in the two middle levels of the Boolean lattice B_(2k+1)</a>, University of Puerto Rico, 2018.
%H A009766 Italo J. Dejter, <a href="https://arxiv.org/abs/1911.02100">Reinterpreting Mütze's Theorem via Natural Enumeration of Ordered Rooted Trees</a>, arXiv:1911.02100 [math.CO], 2019.
%H A009766 Italo J. Dejter, <a href="https://doi.org/10.5614/ejgta.2021.9.1.13">A numeral system for the middle-levels graphs</a>, Elec. J. Graph Theory and Applications (2021) Vol. 9, No. 1, 137-156. See p. 139.
%H A009766 B. Derrida, E. Domany, and D. Mukamel, <a href="http://dx.doi.org/10.1007/BF01050430">An exact solution of a one-dimensional asymmetric exclusion model with open boundaries</a>, J. Stat. Phys. 69, 1992, 667-687; eqs. (20), (21), p. 672. (Y_{N}(K) = A009766(N+1,K-1), 1 <= K <= N+1, N >=0 if alpha = 1 = beta).
%H A009766 E. Deutsch and L. Shapiro, <a href="http://dx.doi.org/10.1016/S0012-365X(01)00121-2">A survey of the Fine numbers</a>, Discrete Math., 241 (2001), 241-265.
%H A009766 Filippo Disanto, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Disanto/disanto5.html">Some Statistics on the Hypercubes of Catalan Permutations</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.2.
%H A009766 Paul Drube, <a href="http://arxiv.org/abs/1606.04869">Generating Functions for Inverted Semistandard Young Tableaux and Generalized Ballot Numbers</a>, arXiv:1606.04869 [math.CO], 2016.
%H A009766 Richard Ehrenborg, Gábor Hetyei, and Margaret Readdy, <a href="https://arxiv.org/abs/1901.07113">Classification of uniform flag triangulations of the Legendre polytope</a>, arXiv:1901.07113 [math.CO], 2019.
%H A009766 R. Ehrenborg, S. Kitaev, and E. Steingrimsson, <a href="http://arxiv.org/abs/1310.1520">Number of cycles in the graph of 312-avoiding permutations</a>, arXiv:1310.1520 [math.CO], 2013.
%H A009766 W. J. R. Eplett, <a href="http://dx.doi.org/10.1016/0012-365X(79)90085-2">A note about the Catalan triangle</a>, Discrete Math. 25(1979), no. 3, 289--291. MR0534947 (80i:05007)
%H A009766 Jackson Evoniuk, Steven Klee, and Van Magnan, <a href="https://www.emis.de/journals/JIS/VOL21/Klee/klee2.html">Enumerating Minimal Length Lattice Paths</a>, J. Int. Seq., Vol. 21 (2018), Article 18.3.6.
%H A009766 G. Feinberg and K.-H. Lee, <a href="http://arxiv.org/abs/1401.0845">Homogeneous representations of KLR-algebras and fully commutative elements</a>, arXiv:1401.0845 [math.RT], 2014.
%H A009766 I. Fanti, A. Frosini, E. Grazzini, R. Pinzani, and S. Rinaldi, <a href="http://puma.dimai.unifi.it/18_3_4/FantiFrosiniGrazziniPinzaniRinaldi.pdf">Characterization and enumeration of some classes of permutominoes</a>, PU. M. A., Vol. 18 (2007), No. 3-4, pp. 265-290.
%H A009766 Dominique Foata and Guo-Niu Han, <a href="http://dx.doi.org/10.1007/s11139-009-9194-9">The doubloon polynomial triangle</a>, Ram. J. 23 (2010), 107-126.
%H A009766 Dominique Foata and Guo-Niu Han, <a href="http://dx.doi.org/10.1093/qmath/hap043">Doubloons and new q-tangent numbers</a>, Quart. J. Math. 62 (2) (2011) 417-432.
%H A009766 H. G. Forder, <a href="http://www.jstor.org/stable/3612775">Some problems in combinatorics</a>, Math. Gazette, vol. 45, 1961, 199-201.
%H A009766 C. A. Francisco, J. Mermin, and J. Schweig, <a href="https://www.math.okstate.edu/~jayjs/ppt.pdf">Catalan numbers, binary trees, and pointed pseudotriangulations</a>, preprint 2013; European Journal of Combinatorics, Volume 45, April 2015, Pages 85-96.
%H A009766 Shishuo Fu, Zhicong Lin, and Yaling Wang, <a href="https://arxiv.org/abs/2011.11302">A combinatorial bijection on di-sk trees</a>, arXiv:2011.11302 [math.CO], 2020.
%H A009766 Ling Gao, <a href="http://hdl.handle.net/20.500.12680/h989rb533">Graph assembly for spider and tadpole graphs</a>, Master's Thesis, Cal. State Poly. Univ. (2023). See p. 12.
%H A009766 Ira Gessel, <a href="http://people.brandeis.edu/~gessel/homepage/papers/superballot.pdf">Super ballot numbers</a>.
%H A009766 Samuele Giraudo, <a href="https://arxiv.org/abs/1903.00677">Tree series and pattern avoidance in syntax trees</a>, arXiv:1903.00677 [math.CO], 2019.
%H A009766 Claude Godrèche and Jean-Marc Luck, <a href="https://arxiv.org/abs/2501.17268">On the first positive position of a random walker</a>, arXiv:2501.17268 [cond-mat.stat-mech], 2025.
%H A009766 Niket Gowravaram, <a href="https://math.mit.edu/research/highschool/primes/materials/2015/Niket.pdf">A Variation of the nil-Temperley-Lieb algebras of type A</a>, Preprint, 2015.
%H A009766 J. M. Hammersley, <a href="http://www.appliedprobability.org/data/files/TMS%20articles/14_1_1.pdf">An undergraduate exercise in manipulation</a>, Math. Scientist, 14 (1989), 1-23.
%H A009766 J. M. Hammersley, <a href="/A006846/a006846.pdf">An undergraduate exercise in manipulation</a>, Math. Scientist, 14 (1989), 1-23. (Annotated scanned copy)
%H A009766 Pamela E. Harris, Thiago Holleben, J. Carlos Martínez Mori, Amanda Priestley, Keith Sullivan, and Per Wagenius, <a href="https://arxiv.org/abs/2501.11718">A Probabilistic Parking Process and Labeled IDLA</a>, arXiv:2501.11718 [math.PR], 2025. See pp. 8, 28.
%H A009766 David He and Daniel Tubbenhauer, <a href="https://arxiv.org/abs/2508.04054">Tensor powers of representations of (diagram) monoids</a>, arXiv:2508.04054 [math.RT], 2025. See p. 7.
%H A009766 Nickolas Hein and Jia Huang, <a href="https://arxiv.org/abs/1807.04623">Variations of the Catalan numbers from some nonassociative binary operations</a>, arXiv:1807.04623 [math.CO], 2018.
%H A009766 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 A009766 F. Hivert, J.-C. Novelli, and J.-Y. Thibon, <a href="https://arxiv.org/abs/math/0401089">The Algebra of Binary Search Trees</a>, Theoretical Computer Science, 339 (2005), 129-165.
%H A009766 R. L. Hudson and Y. Pei, <a href="https://www.researchgate.net/publication/278413670">On a causal quantum stochastic double product integral related to Lévy area</a>, Research Gate, 2015.
%H A009766 R. L. Hudson and Y. Pei, <a href="http://arxiv.org/abs/1506.04294">On a quantum causal stochastic double product integral related to Levy area</a>, arXiv:1506.04294 [math-ph], 2015.
%H A009766 Brant Jones, <a href="https://arxiv.org/abs/1812.00963">Avoiding patterns and making the best choice</a>, arXiv:1812.00963 [math.CO], 2018.
%H A009766 A. Karttunen, <a href="http://oeis.org/wiki/User:Antti_Karttunen">Some notes on Catalan's Triangle</a>.
%H A009766 Lord C. Kavi and Michael W. Newman, <a href="https://arxiv.org/abs/2212.08795">Counting closed walks in infinite regular trees using Catalan and Borel's triangles</a>, arXiv:2212.08795 [math.CO], 2022.
%H A009766 Dongsu Kim and Zhicong Lin, <a href="https://arxiv.org/abs/1706.07208">Refined restricted inversion sequences</a>, arXiv:1706.07208 [math.CO], 2017.
%H A009766 W. Krandick, <a href="http://dx.doi.org/10.1016/j.cam.2003.08.018">Trees and jumps and real roots</a>, J. Computational and Applied Math., 162, 2004, 51-55.
%H A009766 G. Kreweras, <a href="http://www.numdam.org/item?id=BURO_1970__15__3_0">Sur les éventails de segments</a>, Cahiers du Bureau Universitaire de Recherche Opérationelle, Cahier no. 15, Paris, 1970, pp. 3-41.
%H A009766 G. Kreweras, <a href="/A000108/a000108_1.pdf">Sur les éventails de segments</a>, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #15 (1970), 3-41. [Annotated scanned copy]
%H A009766 C. Krishnamachary and M. Bheemasena Rao, <a href="/A000108/a000108_10.pdf">Determinants whose elements are Eulerian, prepared Bernoullian and other numbers</a>, J. Indian Math. Soc., 14 (1922), 55-62, 122-138 and 143-146. [Annotated scanned copy]
%H A009766 A. Laradji and A. Umar, <a href="http://dx.doi.org/10.1007/s00233-004-0101-9">On certain finite semigroups of order-decreasing transformations I</a>, Semigroup Forum 69 (2004), 184-200.
%H A009766 Jeong-Yup Lee, Dong-il Lee, and Sungsoon Kim, <a href="https://arxiv.org/abs/1808.06523">Gröbner-Shirshov bases for Temperley-Lieb algebras of the complex reflection group of type G(d,1,n)</a>, arXiv:1808.06523 [math.RA], 2018.
%H A009766 Kyu-Hwan Lee and Se-jin Oh, <a href="http://arxiv.org/abs/1601.06685">Catalan triangle numbers and binomial coefficients</a>, arXiv:1601.06685 [math.CO], 2016.
%H A009766 Shao-Hua Liu, <a href="https://doi.org/10.1016/j.disc.2019.05.006">The operators F_i on permutations, 132-avoiding permutations and inversions</a>, Discrete Math., 342 (2019), 2402-2414.
%H A009766 Jiaxi Lu and Yuanzhe Ding, <a href="https://arxiv.org/abs/2106.09471">A skeleton model to enumerate standard puzzle sequences</a>, arXiv:2106.09471 [math.CO], 2021.
%H A009766 D. Merlini et al., <a href="http://dx.doi.org/10.1016/S0166-218X(98)00126-7">Underdiagonal lattice paths with unrestricted steps</a>, Discrete Appl. Math., 91 (1999), 197-213.
%H A009766 D. Merlini, R. Sprugnoli and M. C. Verri, <a href="http://dx.doi.org/10.1006/jcta.2002.3273">The tennis ball problem</a>, J. Combin. Theory, A 99 (2002), 307-344 (Table I).
%H A009766 Andrew Misseldine, <a href="http://arxiv.org/abs/1508.03757">Counting Schur Rings over Cyclic Groups</a>, arXiv:1508.03757 [math.RA], 2015.
%H A009766 J.-C. Novelli and J.-Y. Thibon, <a href="https://arxiv.org/abs/math/0512570">Noncommutative Symmetric Functions and Lagrange Inversion</a>, arXiv:math/0512570 [math.CO], 2005-2006.
%H A009766 M. A. A. Obaid, S. K. Nauman, W. M. Fakieh, and C. M. Ringel, <a href="http://www.math.uni-bielefeld.de/~ringel/opus/jeddah.pdf">The numbers of support-tilting modules for a Dynkin algebra</a>, 2014 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Ringel/ringel22.html">J. Int. Seq. 18 (2015) 15.10.6</a>.
%H A009766 P. Pagacz and M. Wojtylak, <a href="http://arxiv.org/abs/1310.2122">On the spectral properties of a class of H-selfadjoint random matrices and the underlying combinatorics</a>, arXiv:1310.2122 [math.PR], 2013.
%H A009766 R. Parviainen, <a href="https://arxiv.org/abs/math/0607793">Permutations, cycles and the pattern 2-13</a>, arXiv:math/0607793 [math.CO], 2006.
%H A009766 R. Pemantle and M. C. Wilson, <a href="https://doi.org/10.1137/050643866">Twenty Combinatorial Examples of Asymptotics Derived from Multivariate Generating Functions</a>, SIAM Rev., 50 (2008), no. 2, 199-272. See p. 261
%H A009766 Planet Math, <a href="http://planetmath.org/latticepathsandballotnumbers">Lattice Paths and ballot numbers</a>.
%H A009766 L. Pudwell, <a href="http://faculty.valpo.edu/lpudwell/slides/pp2013pudwell.pdf">Avoiding an Ordered Partition of Length 3</a>, 2013.
%H A009766 C. M. Ringel, <a href="http://arxiv.org/abs/1502.06553">The Catalan combinatorics of the hereditary artin algebras</a>, arXiv:1502.06553 [math.RT], 2015.
%H A009766 Gabriel Bravo Rios and Agustin Moreno Cañadas, <a href="https://www.fing.edu.uy/imerl/grupos/gia/pdf/Resumen-GBravo.pdf">Dyck Paths in Representation Theory of Algebras</a>, National University of Colombia (2020).
%H A009766 A. Robertson, D. Saracino, and D. Zeilberger, <a href="https://arxiv.org/abs/math/0203033">Refined restricted permutations</a>, arXiv:math/0203033 [math.CO], 2002.
%H A009766 L. W. Shapiro, <a href="http://dx.doi.org/10.1016/0012-365X(76)90009-1">A Catalan triangle</a>, Discrete Math., 14, 83-90, 1976.
%H A009766 T. Sillke, <a href="http://www.mathematik.uni-bielefeld.de/~sillke/SEQUENCES/series004">Catalan's numbers</a>
%H A009766 Hermann Stamm-Wilbrandt, <a href="/A009766/a009766.png">Visualization for all well-formed strings of 4 parenthesis pairs</a> and relation to T(n,k).
%H A009766 R. A. Sulanke, <a href="http://math.boisestate.edu/~sulanke/recentpapindex.html">Guessing, ballot numbers and refining Pascal's triangle</a>
%H A009766 Yidong Sun, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v17i1n20">A simple bijection between binary trees and colored ternary trees</a>, El. J. Combinat. 17 (2010) #N20
%H A009766 Paweł J. Szabłowski, <a href="https://arxiv.org/abs/2106.10461">Yet another way of calculating moments of the Kesten's distribution and its consequences</a>, arXiv:2106.10461 [math.CO], 2021.
%H A009766 Benjamin Testart, <a href="https://arxiv.org/abs/2407.07701">Completing the enumeration of inversion sequences avoiding one or two patterns of length 3</a>, arXiv:2407.07701 [math.CO], 2024. See pp. 9, 29.
%H A009766 Rafael Vazquez and M. Krstic, <a href="http://arxiv.org/abs/1601.02010">Boundary control of a singular reaction-diffusion equation on a disk</a>, arXiv:1601.02010 [math.OC], 2016.
%H A009766 Luis Verde-Star, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Verde/verde4.html">A Matrix Approach to Generalized Delannoy and Schröder Arrays</a>, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.
%H A009766 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CatalansTriangle.html">Catalan's Triangle</a>
%H A009766 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/NonnegativePartialSum.html">Nonnegative Partial Sum</a>
%H A009766 Tad White, <a href="https://arxiv.org/abs/2401.01462">Quota Trees</a>, arXiv:2401.01462 [math.CO], 2024. See p. 17.
%H A009766 Martha Yip, <a href="https://arxiv.org/abs/1703.00057">Rook Placements and Jordan Forms of Upper-Triangular Nilpotent Matrices</a>, arXiv:1703.00057 [math.CO], 2017; The Electronic Journal of Combinatorics 25(1) (2018), #P1.68.
%F A009766 T(n, m) = binomial(n+m, n)*(n-m+1)/(n+1), 0 <= m <= n.
%F A009766 G.f. for column m: (x^m)*N(2; m-1, x)/(1-x)^(m+1), m >= 0, with the row polynomials from triangle A062991 and N(2; -1, x) := 1.
%F A009766 G.f.: C(t*x)/(1-x*C(t*x)) = 1 + (1+t)*x + (1+2*t+2*t^2)*x^2 + ..., where C(x) = (1-sqrt(1-4*x))/(2*x) is the Catalan function. - _Emeric Deutsch_, May 18 2004
%F A009766 Another version of triangle T(n, k) given by [1, 0, 0, 0, 0, 0, ...] DELTA [0, 1, 1, 1, 1, 1, 1, ...] = 1; 1, 0; 1, 1, 0; 1, 2, 2, 0; 1, 3, 5, 5, 0; 1, 4, 9, 14, 14, 0; ... where DELTA is the operator defined in A084938. - _Philippe Deléham_, Feb 16 2005
%F A009766 O.g.f. (with offset 1) is the series reversion of x*(1+x*(1-t))/(1+x)^2 = x - x^2*(1+t) + x^3*(1+2*t) - x^4*(1+3*t) + ... . - _Peter Bala_, Jul 15 2012
%F A009766 Sum_{k=0..floor(n/2)} T(n-k+p-1, k+p-1) = A001405(n+2*p-2) - C(n+2*p-2, p-2), p >= 1. - _Johannes W. Meijer_, Oct 03 2013
%F A009766 Let A(x,t) denote the o.g.f. Then 1 + x*d/dx(A(x,t))/A(x,t) = 1 + (1 + t)*x + (1 + 2*t + 3*t^2)*x^2 + (1 + 3*t + 6*t^2 + 10*t^3)*x^3 + ... is the o.g.f. for A059481. - _Peter Bala_, Jul 21 2015
%F A009766 The n-th row polynomial equals the n-th degree Taylor polynomial of the function (1 - 2*x)/(1 - x)^(n+2) about 0. For example, for n = 4, (1 - 2*x)/(1 - x)^6 = 1 + 4*x + 9*x^2 + 14*x^3 + 14*x^4 + O(x^5). - _Peter Bala_, Feb 18 2018
%F A009766 T(n,k) = binomial(n + k + 1, k) - 2*binomial(n + k, k - 1), for 0 <= k <= n. - _David Callan_, Jun 15 2022
%e A009766 Triangle begins in row n=0 with 0 <= k <= n:
%e A009766 1;
%e A009766 1, 1;
%e A009766 1, 2, 2;
%e A009766 1, 3, 5, 5;
%e A009766 1, 4, 9, 14, 14;
%e A009766 1, 5, 14, 28, 42, 42;
%e A009766 1, 6, 20, 48, 90, 132, 132;
%e A009766 1, 7, 27, 75, 165, 297, 429, 429;
%e A009766 1, 8, 35, 110, 275, 572, 1001, 1430, 1430;
%e A009766 1, 9, 44, 154, 429, 1001, 2002, 3432, 4862, 4862;
%p A009766 A009766 := proc(n,k) binomial(n+k,n)*(n-k+1)/(n+1); end proc:
%p A009766 seq(seq(A009766(n,k), k=0..n), n=0..10); # _R. J. Mathar_, Dec 03 2010
%t A009766 Flatten[NestList[Append[Accumulate[#], Last[Accumulate[#]]] &, {1}, 9]] (* _Birkas Gyorgy_, May 19 2012 *)
%t A009766 T[n_, k_] := T[n, k] = Which[k == 0, 1, k>n, 0, True, T[n-1, k] + T[n, k-1] ]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Mar 07 2016 *)
%o A009766 (PARI) {T(n, k) = if( k<0 || k>n, 0, binomial(n+1+k, k) * (n+1-k) / (n+1+k) )}; /* _Michael Somos_, Oct 17 2006 */
%o A009766 (PARI) b009766=(n1=0,n2=100)->{my(q=if(!n1,0,binomial(n1+1,2)));for(w=n1,n2,for(k=0,w,write("b009766.txt",1*q" "1*(binomial(w+k,w)-binomial(w+k,w+1)));q++))} \\ _R. J. Cano_, Jul 22 2014
%o A009766 (Haskell)
%o A009766 a009766 n k = a009766_tabl !! n !! k
%o A009766 a009766_row n = a009766_tabl !! n
%o A009766 a009766_tabl = iterate (\row -> scanl1 (+) (row ++ [0])) [1]
%o A009766 -- _Reinhard Zumkeller_, Jul 12 2012
%o A009766 (Sage)
%o A009766 @CachedFunction
%o A009766 def ballot(p,q):
%o A009766 if p == 0 and q == 0: return 1
%o A009766 if p < 0 or p > q: return 0
%o A009766 S = ballot(p-2, q) + ballot(p, q-2)
%o A009766 if q % 2 == 1: S += ballot(p-1, q-1)
%o A009766 return S
%o A009766 A009766 = lambda n, k: ballot(2*k, 2*n)
%o A009766 for n in (0..7): [A009766(n, k) for k in (0..n)] # _Peter Luschny_, Mar 05 2014
%o A009766 (Sage) [[binomial(n+k,n)*(n-k+1)/(n+1) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Mar 07 2019
%o A009766 (GAP) Flat(List([0..10],n->List([0..n],m->Binomial(n+m,n)*(n-m+1)/(n+1)))); # _Muniru A Asiru_, Feb 18 2018
%o A009766 (Magma) [[Binomial(n+k,n)*(n-k+1)/(n+1): k in [0..n]]: n in [0..10]]; // _G. C. Greubel_, Mar 07 2019
%o A009766 (Python)
%o A009766 from math import comb, isqrt
%o A009766 def A009766(n): return comb((a:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)))+(b:=n-comb(a+1,2)),b)*(a-b+1)//(a+1) # _Chai Wah Wu_, Nov 12 2024
%Y A009766 The following are all versions of (essentially) the same Catalan triangle: A009766, A008315, A028364, A030237, A047072, A053121, A059365, A062103, A099039, A106566, A130020, A140344.
%Y A009766 Cf. A062745, A214292.
%Y A009766 Sums of the shallow diagonals give A001405, central binomial coefficients: 1=1, 1=1, 1+1=2, 1+2=3, 1+3+2=6, 1+4+5=10, 1+5+9+5=20, ...
%Y A009766 Row sums as well as the sums of the squares of the shallow diagonals give Catalan numbers (A000108).
%Y A009766 Reflected version of A033184.
%Y A009766 Diagonals give A000108, A000108, A000245, A002057, A000344, A003517, A000588, A003518, A003519, A001392, ...
%Y A009766 Triangle sums (see the comments): A000108 (Row1), A000957 (Row2), A001405 (Kn11), A014495 (Kn12), A194124 (Kn13), A030238 (Kn21), A000984 (Kn4), A000958 (Fi2), A165407 (Ca1), A026726 (Ca4), A081696 (Ze2).
%K A009766 nonn,tabl,nice
%O A009766 0,5
%A A009766 _Wouter Meeussen_