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 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). 105

%I

%S 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 1,7,27,75,165,297,429,429,1,8,35,110,275,572,1001,1430,1430,1,9,44,

%U 154,429,1001,2002,3432,4862,4862,1,10,54,208,637,1638,3640,7072,11934

%N 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 The entries in this triangle (in its many forms) are often called ballot numbers.

%C 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 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 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 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 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 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 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 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 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

%D William Feller, "Introduction to Probability Theory and its Applications", vol. I, ed. 2, chap.3, sect.1,2.

%D 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 D. E. Knuth, TAOCP, Vol. 4, Section 7.2.1.6, Eq. 22, p. 451.

%D 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 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).

%H T. D. Noe, <a href="/A009766/b009766.txt">Rows n = 0..100 of triangle, flattened</a>

%H Erik Aas, Arvind Ayyer, Svante Linusson, 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 Ron M. Adin, E. Bagno, 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 Kassie Archer, Abigail Bishop, Alexander Diaz-Lopez, Luis David Garcia Puente, Darren Glass, Joel Louwsma, <a href="https://arxiv.org/abs/1903.01393">Arithmetical structures on bidents</a>, arXiv:1903.01393 [math.CO], 2019.

%H 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 Jean-Christophe Aval, <a href="http://arxiv.org/abs/0711.0906">Multivariate Fuss-Catalan numbers</a>, arXiv:0711.0906 [math.CO], 2007.

%H 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 Axel Bacher, Antonio Bernini, Luca Ferrari, Benjamin Gunby, Renzo Pinzani, 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 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 Elena Barcucci, Alberto Del Lungo, Elisa Pergola, 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 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 P. 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 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 P. Barry, 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 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 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 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 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 Benjamin Braun, Hugo Corrales, Scott Corry, Luis David García Puente, Darren Glass, Nathan Kaplan, Jeremy L. Martin, Gregg Musiker, 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 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 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 Steve Butler, R. Graham, 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 R. J. Cano, <a href="http://oeis.org/w/images/b/bc/CatalanBooks.pdf">Catalan's books</a>

%H L. Carlitz, <a href="http://www.fq.math.ca/Scanned/10-5/carlitz7.pdf">Sequences, paths, ballot numbers</a>

%H I. 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 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 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 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 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 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 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 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 Richard Ehrenborg, Gábor Hetyei, 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 R. Ehrenborg, S. Kitaev, 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 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 Jackson Evoniuk, Steven Klee, 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 G. Feinberg, 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 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 D. Foata, G-N. 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 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 H. G. Forder, <a href="http://www.jstor.org/stable/3612775">Some problems in combinatorics</a>, Math. Gazette, vol. 45, 1961, 199-201.

%H C. A. Francisco, J. Mermin, 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 Ira Gessel, <a href="http://people.brandeis.edu/~gessel/homepage/papers/superballot.pdf">Super ballot numbers</a>.

%H 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 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 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 J. M. Hammersley, <a href="/A006846/a006846.pdf">An undergraduate exercise in manipulation</a>, Math. Scientist, 14 (1989), 1-23. (Annotated scanned copy)

%H Nickolas Hein, 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 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 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 R. L. Hudson, 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 R. L. Hudson, 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 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 A. Karttunen, <a href="http://oeis.org/wiki/User:Antti_Karttunen">Some notes on Catalan's Triangle</a>.

%H 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 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 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 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 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 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 Jeong-Yup Lee, Dong-il Lee, 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 Kyu-Hwan Lee, 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 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 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 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 Andrew Misseldine, <a href="http://arxiv.org/abs/1508.03757">Counting Schur Rings over Cyclic Groups</a>, arXiv:1508.03757 [math.RA], 2015.

%H 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 M. A. A. Obaid, S. K. Nauman, W. M. Fakieh, 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 P. Pagacz, 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 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 R. Pemantle and M. C. Wilson, <a href="http://dx.doi.org/10.1137/050643866">Twenty Combinatorial Examples of Asymptotics Derived from Multivariate Generating Functions, SIAM Rev., 50 (2008), no. 2, 199-272. See p. 261

%H Planet Math, <a href="http://planetmath.org/latticepathsandballotnumbers">Lattice Paths and ballot numbers</a>.

%H L. Pudwell, <a href="http://faculty.valpo.edu/lpudwell/slides/pp2013pudwell.pdf">Avoiding an Ordered Partition of Length 3</a>, 2013.

%H 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 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 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 T. Sillke, <a href="http://www.mathematik.uni-bielefeld.de/~sillke/SEQUENCES/series004">Catalan's numbers</a>

%H R. A. Sulanke, <a href="http://math.boisestate.edu/~sulanke/recentpapindex.html">Guessing, ballot numbers and refining Pascal's triangle</a>

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

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

%H 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 a(n, m) = binomial(n+m, n)*(n-m+1)/(n+1), 0 <= m <= n.

%F 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 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 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 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 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 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 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

%e Triangle begins in row n=0 with 0 <= k <= n:

%e 1;

%e 1, 1;

%e 1, 2, 2;

%e 1, 3, 5, 5;

%e 1, 4, 9, 14, 14;

%e 1, 5, 14, 28, 42, 42;

%e 1, 6, 20, 48, 90, 132, 132;

%e 1, 7, 27, 75, 165, 297, 429, 429;

%e 1, 8, 35, 110, 275, 572, 1001, 1430, 1430;

%e 1, 9, 44, 154, 429, 1001, 2002, 3432, 4862, 4862;

%p A009766 := proc(n,k) binomial(n+k,n)*(n-k+1)/(n+1); end proc:

%p seq(seq(A009766(n,k), k=0..n), n=0..10); # _R. J. Mathar_, Dec 03 2010

%t Flatten[NestList[Append[Accumulate[#], Last[Accumulate[#]]] &, {1}, 9]] (* _Birkas Gyorgy_, May 19 2012 *)

%t 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 (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 (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 n k = a009766_tabl !! n !! k

%o a009766_row n = a009766_tabl !! n

%o a009766_tabl = iterate (\row -> scanl1 (+) (row ++ )) 

%o -- _Reinhard Zumkeller_, Jul 12 2012

%o (Sage)

%o @CachedFunction

%o def ballot(p,q):

%o if p == 0 and q == 0: return 1

%o if p < 0 or p > q: return 0

%o S = ballot(p-2, q) + ballot(p, q-2)

%o if q % 2 == 1: S += ballot(p-1, q-1)

%o return S

%o A009766 = lambda n, k: ballot(2*k, 2*n)

%o for n in (0..7): [A009766(n, k) for k in (0..n)] # _Peter Luschny_, Mar 05 2014

%o (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 (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 (MAGMA) [[Binomial(n+k,n)*(n-k+1)/(n+1): k in [0..n]]: n in [0..10]]; // _G. C. Greubel_, Mar 07 2019

%Y The following are all versions of (essentially) the same Catalan triangle: A009766, A008315, A028364, A030237, A047072, A053121, A059365, A062103, A099039, A106566, A130020, A140344.

%Y Cf. A062745, A214292.

%Y 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 Row sums as well as the sums of the squares of the shallow diagonals give Catalan numbers (A000108).

%Y Reflected version of A033184.

%Y Diagonals give A000108 A000245 A002057 A000344 A003517 A000588 A003518 A003519 A001392, ...

%Y 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 nonn,tabl,nice,changed

%O 0,5

%A _Wouter Meeussen_

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Last modified October 22 04:25 EDT 2019. Contains 328315 sequences. (Running on oeis4.)