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 A001263 #707 Jul 23 2025 10:04:03
%S A001263 1,1,1,1,3,1,1,6,6,1,1,10,20,10,1,1,15,50,50,15,1,1,21,105,175,105,21,
%T A001263 1,1,28,196,490,490,196,28,1,1,36,336,1176,1764,1176,336,36,1,1,45,
%U A001263 540,2520,5292,5292,2520,540,45,1,1,55,825,4950,13860,19404,13860,4950,825
%N A001263 Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan triangle.
%C A001263 Number of antichains (or order ideals) in the poset 2*(k-1)*(n-k) or plane partitions with rows <= k-1, columns <= n-k and entries <= 2. - _Mitch Harris_, Jul 15 2000
%C A001263 T(n,k) is the number of Dyck n-paths with exactly k peaks. a(n,k) = number of pairs (P,Q) of lattice paths from (0,0) to (k,n+1-k), each consisting of unit steps East or North, such that P lies strictly above Q except at the endpoints. - _David Callan_, Mar 23 2004
%C A001263 Number of permutations of [n] which avoid-132 and have k-1 descents. - _Mike Zabrocki_, Aug 26 2004
%C A001263 T(n,k) is the number of paths through n panes of glass, entering and leaving from one side, of length 2n with k reflections (where traversing one pane of glass is the unit length). - _Mitch Harris_, Jul 06 2006
%C A001263 Antidiagonal sums given by A004148 (without first term).
%C A001263 T(n,k) is the number of full binary trees with n internal nodes and k-1 jumps. 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. - _Emeric Deutsch_, Jan 18 2007
%C A001263 From _Gary W. Adamson_, Oct 22 2007: (Start)
%C A001263 The n-th row can be generated by the following operation using an ascending row of (n-1) triangular terms, (A) and a descending row, (B); e.g., row 6:
%C A001263 A: 1....3....6....10....15
%C A001263 B: 15...10....6.....3.....1
%C A001263 C: 1...15...50....50....15....1 = row 6.
%C A001263 Leftmost column of A,B -> first two terms of C; then followed by the operation B*C/A of current column = next term of row C, (e.g., 10*15/3 = 50). Continuing with the operation, we get row 6: (1, 15, 50, 50, 15, 1). (End)
%C A001263 The previous comment can be upgraded to: The ConvOffsStoT transform of the triangular series; and by rows, row 6 is the ConvOffs transform of (1, 3, 6, 10, 15). Refer to triangle A117401 as another example of the ConvOffsStoT transform, and OEIS under Maple Transforms. - _Gary W. Adamson_, Jul 09 2012
%C A001263 For a connection to Lagrange inversion, see A134264. - _Tom Copeland_, Aug 15 2008
%C A001263 T(n,k) is also the number of order-decreasing and order-preserving mappings (of an n-element set) of height k (height of a mapping is the cardinal of its image set). - _Abdullahi Umar_, Aug 21 2008
%C A001263 Row n of this triangle is the h-vector of the simplicial complex dual to an associahedron of type A_n [Fomin & Reading, p.60]. See A033282 for the corresponding array of f-vectors for associahedra of type A_n. See A008459 and A145903 for the h-vectors for associahedra of type B and type D respectively. The Hilbert transform of this triangle (see A145905 for the definition of this transform) is A145904. - _Peter Bala_, Oct 27 2008
%C A001263 T(n,k) is also the number of noncrossing set partitions of [n] into k blocks. Given a partition P of the set {1,2,...,n}, a crossing in P are four integers [a, b, c, d] with 1 <= a < b < c < d <= n for which a, c are together in a block, and b, d are together in a different block. A noncrossing partition is a partition with no crossings. - _Peter Luschny_, Apr 29 2011
%C A001263 Noncrossing set partitions are also called genus 0 partitions. In terms of genus-dependent Stirling numbers of the second kind S2(n,k,g) that count partitions of genus g of an n-set into k nonempty subsets, one has T(n,k) = S2(n,k,0). - _Robert Coquereaux_, Feb 15 2024
%C A001263 Diagonals of A089732 are rows of A001263. - _Tom Copeland_, May 14 2012
%C A001263 From _Peter Bala_, Aug 07 2013: (Start)
%C A001263 Let E(y) = Sum_{n >= 0} y^n/(n!*(n+1)!) = 1/sqrt(y)*BesselI(1,2*sqrt(y)). Then this triangle is the generalized Riordan array (E(y), y) with respect to the sequence n!*(n+1)! as defined in Wang and Wang.
%C A001263 Generating function E(y)*E(x*y) = 1 + (1 + x)*y/(1!*2!) + (1 + 3*x + x^2)*y^2/(2!*3!) + (1 + 6*x + 6*x^2 + x^3)*y^3/(3!*4!) + .... Cf. A105278 with a generating function exp(y)*E(x*y).
%C A001263 The n-th power of this array has a generating function E(y)^n*E(x*y). In particular, the matrix inverse A103364 has a generating function E(x*y)/E(y). (End)
%C A001263 T(n,k) is the number of nonintersecting n arches above the x axis, starting and ending on vertices 1 to 2n, with k being the number of arches starting on an odd vertice and ending on a higher even vertice. Example: T(3,2)=3 [16,25,34] [14,23,56] [12,36,45]. - _Roger Ford_, Jun 14 2014
%C A001263 Fomin and Reading on p. 31 state that the rows of the Narayana matrix are the h-vectors of the associahedra as well as its dual. - _Tom Copeland_, Jun 27 2017
%C A001263 The row polynomials P(n, x) = Sum_{k=1..n} T(n, k)*x^(k-1), together with P(0, x) = 1, multiplied by (n+1) are the numerator polynomials of the o.g.f.s of the diagonal sequences of the triangle A103371: G(n, x) = (n+1)*P(n, x)/(1 - x)^{2*n+1}, for n >= 0. This is proved with Lagrange's theorem applied to the Riordan triangle A135278 = (1/(1 - x)^2, x/(1 - x)). See an example below. - _Wolfdieter Lang_, Jul 31 2017
%C A001263 T(n,k) is the number of Dyck paths of semilength n with k-1 uu-blocks (pairs of consecutive up-steps). - _Alexander Burstein_, Jun 22 2020
%C A001263 In case you were searching for Narayama numbers, the correct spelling is Narayana. - _N. J. A. Sloane_, Nov 11 2020
%C A001263 Named after the Canadian mathematician Tadepalli Venkata Narayana (1930-1987). They were also called "Runyon numbers" after John P. Runyon (1922-2013) of Bell Telephone Laboratories, who used them in a study of a telephone traffic system. - _Amiram Eldar_, Apr 15 2021 The Narayana numbers were first studied by Percy Alexander MacMahon (see reference, Article 495) as pointed out by Bóna and Sagan (see link). - _Peter Luschny_, Apr 28 2022
%C A001263 From _Andrea Arlette España_, Nov 14 2022: (Start)
%C A001263 T(n,k) is the degree distribution of the paths towards synchronization in the transition diagram associated with the Laplacian system over the complete graph K_n, corresponding to ordered initial conditions x_1 < x_2 < ... < x_n.
%C A001263 T(n,k) for n=2N+1 and k=N+1 is the number of states in the transition diagram associated with the Laplacian system over the complete bipartite graph K_{N,N}, corresponding to ordered (x_1 < x_2 < ... < x_N and x_{N+1} < x_{N+2} < ... < x_{2N}) and balanced (Sum_{i=1..N} x_i/N = Sum_{i=N+1..2N} x_i/N) initial conditions. (End)
%C A001263 From _Gus Wiseman_, Jan 23 2023: (Start)
%C A001263 Also the number of unlabeled ordered rooted trees with n nodes and k leaves. See the link by Marko Riedel. For example, row n = 5 counts the following trees:
%C A001263 ((((o)))) (((o))o) ((o)oo) (oooo)
%C A001263 (((o)o)) ((oo)o)
%C A001263 (((oo))) ((ooo))
%C A001263 ((o)(o)) (o(o)o)
%C A001263 ((o(o))) (o(oo))
%C A001263 (o((o))) (oo(o))
%C A001263 The unordered version is A055277. Leaves in standard ordered trees are counted by A358371. (End)
%D A001263 Berman and Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), pp. 103-124.
%D A001263 Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 196.
%D A001263 P. A. MacMahon, Combinatory Analysis, Vols. 1 and 2, Cambridge University Press, 1915, 1916; reprinted by Chelsea, 1960, Sect. 495.
%D A001263 T. V. Narayana, Lattice Path Combinatorics with Statistical Applications. Univ. Toronto Press, 1979, pp. 100-101.
%D A001263 A. Nkwanta, Lattice paths and RNA secondary structures, in African Americans in Mathematics, ed. N. Dean, Amer. Math. Soc., 1997, pp. 137-147.
%D A001263 T. K. Petersen, Eulerian Numbers, Birkhäuser, 2015, Chapter 2.
%D A001263 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 17.
%D A001263 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 6.36(a) and (b).
%H A001263 T. D. Noe, <a href="/A001263/b001263.txt">Rows n=1..100 of triangle, flattened</a>
%H A001263 M. Aigner, <a href="http://dx.doi.org/10.1016/j.disc.2007.06.012">Enumeration via ballot numbers</a>, Discrete Math., 308 (2008), 2544-2563.
%H A001263 Per Alexandersson, Svante Linusson, Samu Potka, and Joakim Uhlin, <a href="https://arxiv.org/abs/2010.11157">Refined Catalan and Narayana cyclic sieving</a>, arXiv:2010.11157 [math.CO], 2020.
%H A001263 N. Alexeev and A. Tikhomirov, <a href="http://arxiv.org/abs/1501.04615">Singular Values Distribution of Squares of Elliptic Random Matrices and type-B Narayana Polynomials</a>, arXiv preprint arXiv:1501.04615 [math.PR], 2015.
%H A001263 Jarod Alper and Rowan Rowlands, <a href="https://arxiv.org/abs/1709.09286">Syzygies of the apolar ideals of the determinant and permanent</a>, arXiv:1709.09286 [math.AC], 2017.
%H A001263 C. Athanasiadis and C. Savvidou, <a href="http://arxiv.org/abs/1204.0362">The local h-vector of the cluster subdivision of a simplex</a>, arXiv:1204.0362 [math.CO], 2012, p. 8, Lemma 2.4 A_n. [_Tom Copeland_, Oct 19 2014]
%H A001263 Arvind Ayyer, Matthieu Josuat-Vergès, and Sanjay Ramassamy, <a href="https://arxiv.org/abs/1803.10351">Extensions of partial cyclic orders and consecutive coordinate polytopes</a>, arXiv:1803.10351 [math.CO], 2018.
%H A001263 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 A001263 Jean Luc Baril, Rigoberto Flórez, and José L. Ramirez, <a href="http://jl.baril.u-bourgogne.fr/narayana.pdf">Generalized Narayana arrays, restricted Dyck paths, and related bijections</a>, Univ. Bourgogne (France, 2025). See p. 2.
%H A001263 Jean-Luc Baril, Pamela E. Harris, Kimberly J. Harry, Matt McClinton, and José L. Ramírez, <a href="https://arxiv.org/abs/2404.05672">Enumerating runs, valleys, and peaks in Catalan words</a>, arXiv:2404.05672 [math.CO], 2024. See p. 5.
%H A001263 Jean-Luc Baril and Sergey Kirgizov, <a href="http://jl.baril.u-bourgogne.fr/Stirling.pdf">The pure descent statistic on permutations</a>, Preprint, 2016.
%H A001263 M. Barnabei, F. Bonetti, and M. Silmbani, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Silimbani/silimbani3.html">Two Permutation Classes Enumerated by the Central Binomial Coefficients</a>, J. Int. Seq. 16 (2013) #13.3.8.
%H A001263 Marilena Barnabei, Flavio Bonetti, and Niccolò Castronuovo, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Barnabei/barnabei5.html">Motzkin and Catalan Tunnel Polynomials</a>, J. Int. Seq., Vol. 21 (2018), Article 18.8.8.
%H A001263 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Barry/barry91.html">On Integer-Sequence-Based Constructions of Generalized Pascal Triangles</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
%H A001263 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Barry4/barry142.html">On a Generalization of the Narayana Triangle</a>, J. Int. Seq. 14 (2011) # 11.4.5.
%H A001263 Paul Barry, <a href="http://dx.doi.org/10.1016/j.laa.2015.10.032">Riordan arrays, generalized Narayana triangles, and series reversion</a>, Linear Algebra and its Applications, 491 (2016) 343-385.
%H A001263 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Barry/barry321.html">Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices</a>, Journal of Integer Sequences, 19, 2016, #16.3.5.
%H A001263 Paul Barry, <a href="https://arxiv.org/abs/1803.06408">Three Études on a sequence transformation pipeline</a>, arXiv:1803.06408 [math.CO], 2018.
%H A001263 Paul Barry, <a href="https://arxiv.org/abs/1804.05027">The Gamma-Vectors of Pascal-like Triangles Defined by Riordan Arrays</a>, arXiv:1804.05027 [math.CO], 2018.
%H A001263 Paul Barry, <a href="https://arxiv.org/abs/1805.02274">On the f-Matrices of Pascal-like Triangles Defined by Riordan Arrays</a>, arXiv:1805.02274 [math.CO], 2018.
%H A001263 Paul Barry, <a href="https://arxiv.org/abs/2004.04577">On a Central Transform of Integer Sequences</a>, arXiv:2004.04577 [math.CO], 2020.
%H A001263 Paul Barry, <a href="https://arxiv.org/abs/2101.06713">On the inversion of Riordan arrays</a>, arXiv:2101.06713 [math.CO], 2021.
%H A001263 Paul Barry and A. Hennessy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Barry2/barry126.html">A Note on Narayana Triangles and Related Polynomials, Riordan Arrays, and MIMO Capacity Calculations</a>, J. Int. Seq. 14 (2011), Article 11.3.8.
%H A001263 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), Article 12.4.8.
%H A001263 C. Bean, M. Tannock, and H. Ulfarsson, <a href="http://arxiv.org/abs/1512.08155">Pattern avoiding permutations and independent sets in graphs</a>, arXiv:1512.08155 [math.CO], 2015.
%H A001263 S. Benchekroun and P. Moszkowski, <a href="http://dx.doi.org/10.1016/S0012-365X(96)00020-9">A bijective proof of an enumerative property of legal bracketings</a> Discrete Math. 176 (1997), no. 1-3, 273-277.
%H A001263 Carl M. Bender and Gerald V. Dunne, <a href="http://dx.doi.org/10.1063/1.527869">Polynomials and operator orderings</a>, J. Math. Phys. 29 (1988), 1727-1731.
%H A001263 J. Berman and P. Koehler, <a href="/A006356/a006356.pdf">Cardinalities of finite distributive lattices</a>, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy]
%H A001263 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 A001263 Aubrey Blecher, Charlotte Brennan, Arnold Knopfmacher, and Toufik Mansour, <a href="https://doi.org/10.1016/j.disc.2017.06.003">The perimeter of words</a>, Discrete Mathematics, 340, no. 10 (2017): 2456-2465.
%H A001263 Miklós Bóna and Bruce E. Sagan, <a href="https://arxiv.org/abs/math/0505382">On divisibility of Narayana numbers by primes</a>, arXiv:math/0505382 [math.CO], 2005.
%H A001263 Miklós Bóna and Bruce E. Sagan, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL8/Sagan/sagan101.html">On Divisibility of Narayana Numbers by Primes</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.4.
%H A001263 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 A001263 J. M. Borwein, <a href="https://carmamaths.org/resources/jon/beauty.pdf">A short walk can be beautiful</a>, 2015.
%H A001263 Jonathan M. Borwein, Armin Straub, and Christophe Vignat, <a href="http://carmamaths.org/resources/jon/dwalks.pdf">Densities of short uniform random walks, Part II: Higher dimensions</a>, Preprint, 2015.
%H A001263 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. 16.
%H A001263 Kevin Buchin, Man-Kwun Chiu, Stefan Felsner, Günter Rote, and André Schulz, <a href="https://arxiv.org/abs/1903.01095">The Number of Convex Polyominoes with Given Height and Width</a>, arXiv:1903.01095 [math.CO], 2019.
%H A001263 Michael Bukata, Ryan Kulwicki, Nicholas Lewandowski, Lara Pudwell, Jacob Roth, and Teresa Wheeland, <a href="https://arxiv.org/abs/1812.07112">Distributions of Statistics over Pattern-Avoiding Permutations</a>, arXiv:1812.07112 [math.CO], 2018.
%H A001263 Alexander Burstein, <a href="https://arxiv.org/abs/2009.00760">Distribution of peak heights modulo k and double descents on k-Dyck paths</a>, arXiv:2009.00760 [math.CO], 2020.
%H A001263 Steve Butler, Kimberly Hadaway, Victoria Lenius, Preston Martens, and Marshall Moats, <a href="https://arxiv.org/abs/2412.07873">Lucky cars and lucky spots in parking functions</a>, arXiv:2412.07873 [math.CO], 2024. See p. 12.
%H A001263 D. Callan, T. Mansour, and M. Shattuck, <a href="http://arxiv.org/abs/1403.6933">Restricted ascent sequences and Catalan numbers</a>, arXiv:1403.6933 [math.CO], 2014.
%H A001263 L. Carlitz, and John Riordan, <a href="http://dx.doi.org/10.1016/0097-3165(71)90032-X">Enumeration of some two-line arrays by extent</a>. J. Combinatorial Theory Ser. A 10 1971 271--283. MR0274301(43 #66). (Coefficients of the polynomials A_n(z) defined in (3.9)).
%H A001263 Ricky X. F. Chen and Christian M. Reidys, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Chen/chen32.html">A Combinatorial Identity Concerning Plane Colored Trees and its Applications</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.7.
%H A001263 Xi Chen, H. Liang, and Y. Wang, <a href="http://dx.doi.org/10.1016/j.laa.2015.01.009">Total positivity of recursive matrices</a>, Linear Algebra and its Applications, Volume 471, 15 April 2015, Pages 383-393.
%H A001263 Xi Chen, H. Liang, and Y. Wang, <a href="http://arxiv.org/abs/1601.05645">Total positivity of recursive matrices</a>, arXiv:1601.05645 [math.CO], 2016.
%H A001263 Johann Cigler, <a href="http://arxiv.org/abs/1501.04750">Some remarks and conjectures related to lattice paths in strips along the x-axis</a>, arXiv:1501.04750 [math.CO], 2015-2016.
%H A001263 Tom Copeland, <a href="http://tcjpn.wordpress.com/2015/12/21/generators-inversion-and-matrix-binomial-and-integral-transforms/">Generators, Inversion, and Matrix, Binomial, and Integral Transforms</a>, 2015.
%H A001263 Robert Coquereaux and Jean-Bernard Zuber, <a href="https://arxiv.org/abs/2305.01100">Counting partitions by genus. A compendium of results</a>, arXiv:2305.01100 [math.CO], 2023. See p. 7.
%H A001263 Robert Coquereaux and Jean-Bernard Zuber, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL27/Coquereaux/coque5.html">Counting partitions by genus: a compendium of results</a>, Journal of Integer Sequences, Vol. 27 (2024), Article 24.2.6. See p. 12.
%H A001263 R. Cori and G. Hetyei, <a href="http://arxiv.org/abs/1306.4628">Counting genus one partitions and permutations</a>, arXiv:1306.4628 [math.CO], 2013.
%H A001263 R. Cori and G. Hetyei, <a href="https://web.archive.org/web/20160616221055/http://www.dmtcs.org/dmtcs-ojs/index.php/proceedings/article/viewFile/dmAT0130/4488">How to count genus one partitions</a>, FPSAC 2014, Chicago, Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France, 2014, 333-344.
%H A001263 R. De Castro, A. L. Ramírez, and J. L. Ramírez, <a href="http://arxiv.org/abs/1310.2449">Applications in Enumerative Combinatorics of Infinite Weighted Automata and Graphs</a>, arXiv:1310.2449 [cs.DM], 2013.
%H A001263 Colin Defant, <a href="https://arxiv.org/abs/1903.09138">Counting 3-Stack-Sortable Permutations</a>, arXiv:1903.09138 [math.CO], 2019.
%H A001263 Yun Ding and Rosena R. X. Du, <a href="http://arxiv.org/abs/1109.2661">Counting Humps in Motzkin paths</a>, arXiv:1109.2661 [math.CO], 2011.
%H A001263 T. Doslic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Doslic/doslic6.html">Handshakes across a (round) table</a>, JIS 13 (2010) #10.2.7.
%H A001263 T. Doslic, D. Svrtan, and D. Veljan, <a href="http://dx.doi.org/10.1016/j.disc.2004.04.001">Enumerative aspects of secondary structures</a>, Discr. Math., 285 (2004), 67-82.
%H A001263 D. Drake, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Drake/drake.html">Bijections from Weighted Dyck Paths to Schröder Paths</a>, J. Int. Seq. 13 (2010) # 10.9.2.
%H A001263 Jennifer Elder, Nadia Lafrenière, Erin McNicholas, Jessica Striker, and Amanda Welch, <a href="https://arxiv.org/abs/2307.08520">Toggling, rowmotion, and homomesy on interval-closed sets</a>, arXiv:2307.08520 [math.CO], 2023.
%H A001263 Sergi Elizalde, Johnny Rivera Jr., and Yan Zhuang, <a href="https://arxiv.org/abs/2408.15111">Counting pattern-avoiding permutations by big descents</a>, arXiv:2408.15111 [math.CO], 2024. See p. 18.
%H A001263 A. España, X. Leoncini, and E. Ugalde, <a href="https://arxiv.org/abs/2205.05948">Combinatorics of the paths towards synchronization</a>, arXiv:2205.05948 [math.DS], 2022.
%H A001263 L. Ferrari and E. Munarini, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Ferrari/ferrari.html">Enumeration of Edges in Some Lattices of Paths</a>, J. Int. Seq. 17 (2014) #14.1.5.
%H A001263 FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/StatisticsDatabase/St000015">The number of peaks of a Dyck path</a>, <a href="http://www.findstat.org/StatisticsDatabase/St000024">The number of double rises of a Dyck path</a>, <a href="http://www.findstat.org/StatisticsDatabase/St000053">The number of valleys of a Dyck path</a>, <a href="http://www.findstat.org/StatisticsDatabase/St000083">The number of left oriented leafs except the first one of a binary tree</a>, <a href="http://www.findstat.org/StatisticsDatabase/St000120">The number of left tunnels of a Dyck path</a>.
%H A001263 T. A. Fisher, <a href="http://arxiv.org/abs/1510.07484">A Caldero-Chapoton map depending on a torsion class</a>, arXiv:1510.07484 [math.RT], 2015-2016.
%H A001263 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; see page 182.
%H A001263 S. Fomin and N. Reading, <a href="http://arxiv.org/abs/math.CO/0505518">Root systems and generalized associahedra</a>, Lecture notes for IAS/Park-City 2004.
%H A001263 Alessandra Frabetti, <a href="https://doi.org/10.1023/A:1008723801201">Simplicial properties of the set of planar binary trees</a>, Journal of Algebraic Combinatorics 13, 41-65 (2001).
%H A001263 Samuele Giraudo, <a href="http://arxiv.org/abs/1107.3472">Intervals of balanced binary trees in the Tamari lattice</a>, arXiv:1107.3472 [math.CO], 2011.
%H A001263 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 A001263 R. L. Graham and J. Riordan, <a href="http://www.jstor.org/stable/2314791">The solution of a certain recurrence</a>, Amer. Math. Monthly 73, 1966, pp. 604-608.
%H A001263 R. K. Guy, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/GUY/catwalks.html">Catwalks, sandsteps and Pascal pyramids</a>, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
%H A001263 Kevin G. Hare and Ghislain McKay, <a href="http://arxiv.org/abs/1506.01313">Some properties of even moments of uniform random walks</a>, arXiv:1506.01313 [math.CO], 2015.
%H A001263 F. Hivert, J.-C. Novelli, and J.-Y. Thibon, <a href="https://arxiv.org/abs/math/0605262">Commutative combinatorial Hopf algebras</a>, arXiv:math/0605262 [math.CO], 2006.
%H A001263 H. E. Hoggatt, Jr., <a href="http://www.mathstat.dal.ca/FQ/Scanned/12-3/hoggatt.pdf">Triangular Numbers</a>, Fibonacci Quarterly 12 (Oct. 1974), 221-230.
%H A001263 F. K. Hwang and C. L. Mallows, <a href="http://dx.doi.org/10.1016/0097-3165(95)90097-7">Enumerating nested and consecutive partitions</a>, J. Combin. Theory Ser. A 70 (1995), no. 2, 323-333.
%H A001263 M. Hyatt and J. Remmel, <a href="http://arxiv.org/abs/1208.1052">The classification of 231-avoiding permutations by descents and maximum drop</a>, arXiv:1208.1052 [math.CO], 2012.
%H A001263 Matthieu Josuat-Verges, <a href="https://doi.org/10.1007/s11139-017-9885-6">A q-analog of Schläfli and Gould identities on Stirling numbers</a>, Ramanujan J 46, 483-507 (2018); <a href="https://arxiv.org/abs/1610.02965">arXiv preprint</a>, 2016.
%H A001263 S. Kamioka, <a href="http://arxiv.org/abs/1309.0268">Laurent biorthogonal polynomials, q-Narayana polynomials and domino tilings of the Aztec diamonds</a>, arXiv:1309.0268 [math.CO], 2013.
%H A001263 Thomas Koshy, <a href="/A001263/a001263.jpg">Illustration of triangle with dark color for odd number, light for even number</a> [Although the illustration says "Applet", this is simply a plain jpeg file]
%H A001263 Vladimir Kostov and Boris Shapiro, <a href="http://arXiv.org/abs/0804.1028">Narayana numbers and Schur-Szego composition</a>, arXiv:0804.1028 [math.CA], 2008.
%H A001263 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 A001263 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 A001263 G. Kreweras, <a href="http://www.numdam.org/item?id=BURO_1967__10__23_0">Traitement simultané du "Problème de Young" et du "Problème de Simon Newcomb"</a>, Cahiers du Bureau Universitaire de Recherche Opérationnelle. Institut de Statistique, Université de Paris, 10 (1967), 23-31.
%H A001263 G. Kreweras, <a href="/A006542/a006542_1.pdf">Traitement simultané du "Problème de Young" et du "Problème de Simon Newcomb"</a>, Cahiers du Bureau Universitaire de Recherche Opérationnelle. Institut de Statistique, Université de Paris, 10 (1967), 23-31. [Annotated scanned copy]
%H A001263 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 Opérationnelle, Institut de Statistique, Université de Paris, #15 (1970), 3-41.
%H A001263 G. Kreweras, <a href="http://www.numdam.org/item?id=MSH_1976__53__5_0">Les préordres totaux compatibles avec un ordre partiel</a>, Math. Sci. Humaines No. 53 (1976), 5-30.
%H A001263 G. Kreweras, <a href="/A019538/a019538.pdf">Les préordres totaux compatibles avec un ordre partiel</a>, Math. Sci. Humaines No. 53 (1976), 5-30. (Annotated scanned copy)
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%H A001263 Nate Kube and Frank Ruskey, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL8/Ruskey/ruskey99.html">Sequences That Satisfy a(n-a(n))=0</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.5.
%H A001263 A. Laradji and A. Umar, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Umar/um.html">Combinatorial Results for Semigroups of Order-Decreasing Partial Transformations</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.8.
%H A001263 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 A001263 Elżbieta Liszewska and Wojciech Młotkowski, <a href="https://arxiv.org/abs/1907.10725">Some relatives of the Catalan sequence</a>, arXiv:1907.10725 [math.CO], 2019.
%H A001263 Amya Luo, <a href="https://math.dartmouth.edu/theses/undergrad/2024/Luo-thesis.pdf">Pattern Avoidance in Nonnesting Permutations</a>, Undergraduate Thesis, Dartmouth College (2024). See p. 16.
%H A001263 Feihu Liu, Guoce Xin, and Chen Zhang, <a href="https://arxiv.org/abs/2412.18744">Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS</a>, arXiv:2412.18744 [math.CO], 2024. See pp. 26-27.
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%H A001263 Toufik Mansour and Mark Shattuck, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v18i2p34">Pattern-avoiding set partitions and Catalan numbers</a>, Electronic Journal of Combinatorics, 18(2) (2012), #P34.
%H A001263 Toufik Mansour and Gökhan Yıldırım, <a href="https://arxiv.org/abs/1808.05430">Permutations avoiding 312 and another pattern, Chebyshev polynomials and longest increasing subsequences</a>, arXiv:1808.05430 [math.CO], 2018.
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%H A001263 J.-C. Novelli and 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:1403.5962 [math.CO], 2014. See Fig. 4.
%H A001263 Judy-anne Osborn, <a href="https://ajc.maths.uq.edu.au/pdf/48/ajc_v48_p243.pdf">Bi-banded paths, a bijection and the Narayana numbers</a>, Australasian Journal of Combinatorics, Volume 48 (2010), Pages 243-252.
%H A001263 T. K. Petersen, <a href="https://doi.org/10.1007/978-1-4939-3091-3">Chapter 2. Narayana numbers.</a> In: Eulerian Numbers. Birkhäuser Basel, 2015. doi:10.1007/978-1-4939-3091-3.
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%H A001263 Lara Pudwell, <a href="http://permutationpatterns.com/slides/Pudwell.pdf">On the distribution of peaks (and other statistics)</a>, 16th International Conference on Permutation Patterns, Dartmouth College, 2018. [Broken link]
%H A001263 Dun Qiu and Jeffery Remmel, <a href="https://arxiv.org/abs/1804.07087">Patterns in words of ordered set partitions</a>, arXiv:1804.07087 [math.CO], 2018.
%H A001263 Marko Riedel, Mathematics Stack Exchange, <a href="https://math.stackexchange.com/questions/3108340/">Narayana numbers count unlabeled ordered rooted trees on n nodes having k leaves, proof.</a>
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%H A001263 M. Sheppeard, <a href="http://vixra.org/pdf/1208.0242v6.pdf">Constructive motives and scattering</a> 2013 (p. 41). [_Tom Copeland_, Oct 03 2014]
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%H A001263 R. A. Sulanke, <a href="https://web.archive.org/web/20180416202336/http://math.boisestate.edu/~sulanke/PAPERS/narayana3ADL.pdf">Three-dimensional Narayana and Schröder numbers</a>
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%H A001263 Wikipedia, <a href="https://en.wikipedia.org/wiki/Noncrossing_partition">Noncrossing partition</a>
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%H A001263 James J. Y. Zhao, <a href="https://arxiv.org/abs/2108.03590">On the positive zeros of generalized Narayana polynomials related to the Boros-Moll polynomials</a>, arXiv:2108.03590 [math.CO], 2021.
%F A001263 a(n, k) = C(n-1, k-1)*C(n, k-1)/k for k!=0; a(n, 0)=0.
%F A001263 Triangle equals [0, 1, 0, 1, 0, 1, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, ...] where DELTA is Deléham's operator defined in A084938.
%F A001263 0<n, 1<=k<=n a(n, 1) = a(n, n) = 1 a(n, k) = sum(i=1..n-1, sum(r=1..k-1, a(n-1-i, k-r) a(i, r))) + a(n-1, k) a(n, k) = sum(i=1..k-1, binomial(n+i-1, 2k-2)*a(k-1, i)) - _Mike Zabrocki_, Aug 26 2004
%F A001263 T(n, k) = C(n, k)*C(n-1, k-1) - C(n, k-1)*C(n-1, k) (determinant of a 2 X 2 subarray of Pascal's triangle A007318). - _Gerald McGarvey_, Feb 24 2005
%F A001263 T(n, k) = binomial(n-1, k-1)^2 - binomial(n-1, k)*binomial(n-1, k-2). - _David Callan_, Nov 02 2005
%F A001263 a(n,k) = C(n,2) (a(n-1,k)/((n-k)*(n-k+1)) + a(n-1,k-1)/(k*(k-1))) a(n,k) = C(n,k)*C(n,k-1)/n. - _Mitch Harris_, Jul 06 2006
%F A001263 Central column = A000891, (2n)!*(2n+1)! / (n!*(n+1)!)^2. - _Zerinvary Lajos_, Oct 29 2006
%F A001263 G.f.: (1-x*(1+y)-sqrt((1-x*(1+y))^2-4*y*x^2))/(2*x) = Sum_{n>0, k>0} a(n, k)*x^n*y^k.
%F A001263 From _Peter Bala_, Oct 22 2008: (Start)
%F A001263 Relation with Jacobi polynomials of parameter (1,1):
%F A001263 Row n+1 generating polynomial equals 1/(n+1)*x*(1-x)^n*Jacobi_P(n,1,1,(1+x)/(1-x)). It follows that the zeros of the Narayana polynomials are all real and nonpositive, as noted above. O.g.f for column k+2: 1/(k+1) * y^(k+2)/(1-y)^(k+3) * Jacobi_P(k,1,1,(1+y)/(1-y)). Cf. A008459.
%F A001263 T(n+1,k) is the number of walks of n unit steps on the square lattice (i.e., each step in the direction either up (U), down (D), right (R) or left (L)) starting from the origin and finishing at lattice points on the x axis and which remain in the upper half-plane y >= 0 [Guy]. For example, T(4,3) = 6 counts the six walks RRL, LRR, RLR, UDL, URD and RUD, from the origin to the lattice point (1,0), each of 3 steps. Compare with tables A145596 - A145599.
%F A001263 Define a functional I on formal power series of the form f(x) = 1 + ax + bx^2 + ... by the following iterative process. Define inductively f^(1)(x) = f(x) and f^(n+1)(x) = f(x*f^(n)(x)) for n >= 1. Then set I(f(x)) = lim_{n -> infinity} f^(n)(x) in the x-adic topology on the ring of formal power series; the operator I may also be defined by I(f(x)) := 1/x*series reversion of x/f(x).
%F A001263 The o.g.f. for this array is I(1 + t*x + t*x^2 + t*x^3 + ...) = 1 + t*x + (t + t^2)*x^2 + (t + 3*t^2 + t^3)*x^3 + ... = 1/(1 - x*t/(1 - x/(1 - x*t/(1 - x/(1 - ...))))) (as a continued fraction). Cf. A108767, A132081 and A141618. (End)
%F A001263 G.f.: 1/(1-x-xy-x^2y/(1-x-xy-x^2y/(1-... (continued fraction). - _Paul Barry_, Sep 28 2010
%F A001263 E.g.f.: exp((1+y)x)*Bessel_I(1,2*sqrt(y)x)/(sqrt(y)*x). - _Paul Barry_, Sep 28 2010
%F A001263 G.f.: A(x,y) = exp( Sum_{n>=1} [Sum_{k=0..n} C(n,k)^2*y^k] * x^n/n ). - _Paul D. Hanna_, Oct 13 2010
%F A001263 With F(x,t) = (1-(1+t)*x-sqrt(1-2*(1+t)*x+((t-1)*x)^2))/(2*x) an o.g.f. in x for the Narayana polynomials in t, G(x,t) = x/(t+(1+t)*x+x^2) is the compositional inverse in x. Consequently, with H(x,t) = 1/ (dG(x,t)/dx) = (t+(1+t)*x+x^2)^2 / (t-x^2), the n-th Narayana polynomial in t is given by (1/n!)*((H(x,t)*D_x)^n)x evaluated at x=0, i.e., F(x,t) = exp(x*H(u,t)*D_u)u, evaluated at u = 0. Also, dF(x,t)/dx = H(F(x,t),t). - _Tom Copeland_, Sep 04 2011
%F A001263 With offset 0, A001263 = Sum_{j>=0} A132710^j / A010790(j), a normalized Bessel fct. May be represented as the Pascal matrix A007318, n!/[(n-k)!*k!], umbralized with b(n)=A002378(n) for n>0 and b(0)=1: A001263(n,k)= b.(n!)/{b.[(n-k)!]*b.(k!)} where b.(n!) = b(n)*b(n-1)...*b(0), a generalized factorial (see example). - _Tom Copeland_, Sep 21 2011
%F A001263 With F(x,t) = {1-(1-t)*x-sqrt[1-2*(1+t)*x+[(t-1)*x]^2]}/2 a shifted o.g.f. in x for the Narayana polynomials in t, G(x,t)= x/[t-1+1/(1-x)] is the compositional inverse in x. Therefore, with H(x,t)=1/(dG(x,t)/dx)=[t-1+1/(1-x)]^2/{t-[x/(1-x)]^2}, (see A119900), the (n-1)-th Narayana polynomial in t is given by (1/n!)*((H(x,t)*d/dx)^n)x evaluated at x=0, i.e., F(x,t) = exp(x*H(u,t)*d/du) u, evaluated at u = 0. Also, dF(x,t)/dx = H(F(x,t),t). - _Tom Copeland_, Sep 30 2011
%F A001263 T(n,k) = binomial(n-1,k-1)*binomial(n+1,k)-binomial(n,k-1)*binomial(n,k). - _Philippe Deléham_, Nov 05 2011
%F A001263 A166360(n-k) = T(n,k) mod 2. - _Reinhard Zumkeller_, Oct 10 2013
%F A001263 Damped sum of a column, in leading order: lim_{d->0} d^(2k-1) Sum_{N>=k} T(N,k)(1-d)^N=Catalan(n). - _Joachim Wuttke_, Sep 11 2014
%F A001263 Multiplying the n-th column by n! generates the revert of the unsigned Lah numbers, A089231. - _Tom Copeland_, Jan 07 2016
%F A001263 Row polynomials: (x - 1)^(n+1)*(P(n+1,(1 + x)/(x - 1)) - P(n-1,(1 + x)/(x - 1)))/((4*n + 2)), n = 1,2,... and where P(n,x) denotes the n-th Legendre polynomial. - _Peter Bala_, Mar 03 2017
%F A001263 The coefficients of the row polynomials R(n, x) = hypergeom([-n,-n-1], [2], x) generate the triangle based in (0,0). - _Peter Luschny_, Mar 19 2018
%F A001263 Multiplying the n-th diagonal by n!, with the main diagonal n=1, generates the Lah matrix A105278. With G equal to the infinitesimal generator of A132710, the Narayana triangle equals Sum_{n >= 0} G^n/((n+1)!*n!) = (sqrt(G))^(-1) * I_1(2*sqrt(G)), where G^0 is the identity matrix and I_1(x) is the modified Bessel function of the first kind of order 1. (cf. Sep 21 2011 formula also.) - _Tom Copeland_, Sep 23 2020
%F A001263 T(n,k) = T(n,k-1)*C(n-k+2,2)/C(k,2). - _Yuchun Ji_, Dec 21 2020
%F A001263 From _Sergii Voloshyn_, Nov 25 2024: (Start)
%F A001263 G.f.: F(x,y) = (1-x*(1+y)-sqrt((1-x*(1+y))^2-4*y*x^2))/(2*x) is the solution of the differential equation x^3 * d^2(x*F(x,y))/dx^2 = y * d^2(x*F(x,y))/dy^2.
%F A001263 Let E be the operator x*D*D, where D denotes the derivative operator d/dx. Then (1/(n! (1 + n)!)) * E^n(x/(1 - x)) = (row n generating polynomial)/(1 - x)^(2*n+1) = Sum_{k >= 0} C(n-1, k-1)*C(n, k-1)/k*x^k. For example, when n = 4 we have (1/4!/5!)*E^3(x/(1 - x)) = x (1 + 6 x + 6 x^2 + x^3)/(1 - x)^9. (End)
%e A001263 The initial rows of the triangle are:
%e A001263 [1] 1
%e A001263 [2] 1, 1
%e A001263 [3] 1, 3, 1
%e A001263 [4] 1, 6, 6, 1
%e A001263 [5] 1, 10, 20, 10, 1
%e A001263 [6] 1, 15, 50, 50, 15, 1
%e A001263 [7] 1, 21, 105, 175, 105, 21, 1
%e A001263 [8] 1, 28, 196, 490, 490, 196, 28, 1
%e A001263 [9] 1, 36, 336, 1176, 1764, 1176, 336, 36, 1;
%e A001263 ...
%e A001263 For all n, 12...n (1 block) and 1|2|3|...|n (n blocks) are noncrossing set partitions.
%e A001263 Example of umbral representation:
%e A001263 A007318(5,k)=[1,5/1,5*4/(2*1),...,1]=(1,5,10,10,5,1),
%e A001263 so A001263(5,k)={1,b(5)/b(1),b(5)*b(4)/[b(2)*b(1)],...,1}
%e A001263 = [1,30/2,30*20/(6*2),...,1]=(1,15,50,50,15,1).
%e A001263 First = last term = b.(5!)/[b.(0!)*b.(5!)]= 1. - _Tom Copeland_, Sep 21 2011
%e A001263 Row polynomials and diagonal sequences of A103371: n = 4, P(4, x) = 1 + 6*x + 6*x^2 + x^3, and the o.g.f. of fifth diagonal is G(4, x) = 5* P(4, x)/(1 - x)^9, namely [5, 75, 525, ...]. See a comment above. - _Wolfdieter Lang_, Jul 31 2017
%p A001263 A001263 := (n,k)->binomial(n-1,k-1)*binomial(n,k-1)/k;
%p A001263 a:=proc(n,k) option remember; local i; if k=1 or k=n then 1 else add(binomial(n+i-1, 2*k-2)*a(k-1,i),i=1..k-1); fi; end:
%p A001263 # Alternatively, as a (0,0)-based triangle:
%p A001263 R := n -> simplify(hypergeom([-n, -n-1], [2], x)): Trow := n -> seq(coeff(R(n,x),x,j), j=0..n): seq(Trow(n), n=0..9); # _Peter Luschny_, Mar 19 2018
%t A001263 T[n_, k_] := If[k==0, 0, Binomial[n-1, k-1] Binomial[n, k-1] / k];
%t A001263 Flatten[Table[Binomial[n-1,k-1] Binomial[n,k-1]/k,{n,15},{k,n}]] (* _Harvey P. Dale_, Feb 29 2012 *)
%t A001263 TRow[n_] := CoefficientList[Hypergeometric2F1[1 - n, -n, 2, x], x];
%t A001263 Table[TRow[n], {n, 1, 11}] // Flatten (* _Peter Luschny_, Mar 19 2018 *)
%t A001263 aot[n_]:=If[n==1,{{}},Join@@Table[Tuples[aot/@c],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
%t A001263 Table[Length[Select[aot[n],Length[Position[#,{}]]==k&]],{n,2,9},{k,1,n-1}] (* _Gus Wiseman_, Jan 23 2023 *)
%t A001263 T[1, 1] := 1; T[n_, k_]/;1<=k<=n := T[n, k] = (2n/k-1) T[n-1,k-1] + T[n-1, k]; T[n_, k_] := 0; Flatten@Table[T[n, k], {n, 1, 11}, {k, 1, n}] (* _Oliver Seipel_, Dec 31 2024 *)
%o A001263 (PARI) {a(n, k) = if(k==0, 0, binomial(n-1, k-1) * binomial(n, k-1) / k)};
%o A001263 (PARI) {T(n,k)=polcoeff(polcoeff(exp(sum(m=1,n,sum(j=0,m,binomial(m,j)^2*y^j)*x^m/m) +O(x^(n+1))),n,x),k,y)} \\ _Paul D. Hanna_, Oct 13 2010
%o A001263 (Haskell)
%o A001263 a001263 n k = a001263_tabl !! (n-1) !! (k-1)
%o A001263 a001263_row n = a001263_tabl !! (n-1)
%o A001263 a001263_tabl = zipWith dt a007318_tabl (tail a007318_tabl) where
%o A001263 dt us vs = zipWith (-) (zipWith (*) us (tail vs))
%o A001263 (zipWith (*) (tail us ++ [0]) (init vs))
%o A001263 -- _Reinhard Zumkeller_, Oct 10 2013
%o A001263 (Magma) /* triangle */ [[Binomial(n-1,k-1)*Binomial(n,k-1)/k : k in [1..n]]: n in [1.. 15]]; // _Vincenzo Librandi_, Oct 19 2014
%o A001263 (Sage)
%o A001263 @CachedFunction
%o A001263 def T(n, k):
%o A001263 if k == n or k == 1: return 1
%o A001263 if k <= 0 or k > n: return 0
%o A001263 return binomial(n, 2) * (T(n-1, k)/((n-k)*(n-k+1)) + T(n-1, k-1)/(k*(k-1)))
%o A001263 for n in (1..9): print([T(n, k) for k in (1..n)]) # _Peter Luschny_, Oct 28 2014
%o A001263 (GAP) Flat(List([1..11],n->List([1..n],k->Binomial(n-1,k-1)*Binomial(n,k-1)/k))); # _Muniru A Asiru_, Jul 12 2018
%Y A001263 Other versions are in A090181 and A131198. - _Philippe Deléham_, Nov 18 2007
%Y A001263 Cf. variants: A181143, A181144. - _Paul D. Hanna_, Oct 13 2010
%Y A001263 Row sums give A000108 (Catalan numbers), n>0.
%Y A001263 Columns give A000217, A002415, A006542, A006857, A108679. A084938.
%Y A001263 Cf. A000372, A002083, A056932, A056939, A056940, A056941, A065329, A073345.
%Y A001263 A145596, A145597, A145598, A145599. - _Peter Bala_, Oct 22 2008
%Y A001263 A008459 (h-vectors type B associahedra), A033282 (f-vectors type A associahedra), A145903 (h-vectors type D associahedra), A145904 (Hilbert transform). - _Peter Bala_, Oct 27 2008
%Y A001263 Cf. A016098 and A189232 for numbers of crossing set partitions.
%Y A001263 Cf. A243752.
%Y A001263 Cf. A089231, A103371, A135278.
%Y A001263 Cf. A002378, A007318, A010790, A089732, A105278, A119900, A132710, A134264.
%Y A001263 Triangles of generalized binomial coefficients (n,k)_m (or generalized Pascal triangles) for m = 1,...,12: A007318 (Pascal), A001263, A056939, A056940, A056941, A142465, A142467, A142468, A174109, A342889, A342890, A342891.
%Y A001263 Cf. A000081, A005043, A032027, A055277, A358371.
%K A001263 nonn,easy,tabl,nice,look
%O A001263 1,5
%A A001263 _N. J. A. Sloane_
%E A001263 Deleted certain dangerous or potentially dangerous links. - _N. J. A. Sloane_, Jan 30 2021