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%I #624 Apr 14 2024 02:58:41
%S 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 1,1,28,196,490,490,196,28,1,1,36,336,1176,1764,1176,336,36,1,1,45,
%U 540,2520,5292,5292,2520,540,45,1,1,55,825,4950,13860,19404,13860,4950,825
%N 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 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 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 Number of permutations of [n] which avoid-132 and have k-1 descents. - _Mike Zabrocki_, Aug 26 2004
%C 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 Antidiagonal sums given by A004148 (without first term).
%C 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 From _Gary W. Adamson_, Oct 22 2007: (Start)
%C 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 A: 1....3....6....10....15
%C B: 15...10....6.....3.....1
%C C: 1...15...50....50....15....1 = row 6.
%C 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 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 For a connection to Lagrange inversion, see A134264. - _Tom Copeland_, Aug 15 2008
%C 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 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 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 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 Diagonals of A089732 are rows of A001263. - _Tom Copeland_, May 14 2012
%C From _Peter Bala_, Aug 07 2013: (Start)
%C 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 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 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 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 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 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 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 In case you were searching for Narayama numbers, the correct spelling is Narayana. - _N. J. A. Sloane_, Nov 11 2020
%C 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 From _Andrea Arlette España_, Nov 14 2022: (Start)
%C 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 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 From _Gus Wiseman_, Jan 23 2023: (Start)
%C 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 ((((o)))) (((o))o) ((o)oo) (oooo)
%C (((o)o)) ((oo)o)
%C (((oo))) ((ooo))
%C ((o)(o)) (o(o)o)
%C ((o(o))) (o(oo))
%C (o((o))) (oo(o))
%C The unordered version is A055277. Leaves in standard ordered trees are counted by A358371.
%C (End)
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%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 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 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 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 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 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)
%H G. Kreweras, and P. Moszkowski, <a href="http://dx.doi.org/10.1016/0378-3758(86)90011-X">A new enumerative property of the Narayana numbers</a>, Journal of statistical planning and inference 14.1 (1986): 63-67.
%H 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 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 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 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 P. A. MacMahon, <a href="http://www.hti.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABU9009">Combinatory analysis</a>.
%H K. Manes, A. Sapounakis, I. Tasoulas, and P. Tsikouras, <a href="http://dx.doi.org/10.1016/j.disc.2014.07.015">Nonleft peaks in Dyck paths: a combinatorial approach</a>, Discrete Math., 337 (2014), 97-105.
%H Toufik Mansour and Reza Rastegar, <a href="https://arxiv.org/abs/1911.04025">On typical triangulations of a convex n-gon</a>, arXiv:1911.04025 [math.CO], 2019.
%H Toufik Mansour, Reza Rastegar, Alexander Roitershtein, and Gökhan Yıldırım, <a href="https://arxiv.org/abs/2001.10030">The longest increasing subsequence in involutions avoiding 3412 and another pattern</a>, arXiv:2001.10030 [math.CO], 2020.
%H 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 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.
%H A. Marco and J.-J. Martinez, <a href="http://repository.uwyo.edu/ela/vol30/iss1/7">A total positivity property of the Marchenko-Pastur Law</a>, Electronic Journal of Linear Algebra, Volume 30 (2015), #7, pp. 106-117.
%H MathOverflow, <a href="http://mathoverflow.net/questions/273401/narayana-polynomials-as-numerators-of-ehrhart-series-rational-functions">Narayana polynomials as numerator polynomials for Ehrhart series rational functions?</a>, a MO question posed by Tom Copeland and answered by Richard Stanley, 2017.
%H D. Merlini, R. Sprugnoli and M. C. Verri, <a href="http://www.dsi.unifi.it/~merlini/fun01.ps">Waiting patterns for a printer</a>, FUN with algorithm'01, Isola d'Elba, 2001.
%H A. Micheli and D. Rossin, <a href="https://arxiv.org/abs/math/0506538">Edit distance between unlabeled ordered trees</a>, arXiv:math/0506538 [math.CO], 2005.
%H J.-C. Novelli and J.-Y. Thibon, <a href="https://arxiv.org/abs/math/0605061">Polynomial realizations of some trialgebras</a>, arXiv:math/0605061 [math.CO], 2006; Proc. Formal Power Series and Algebraic Combinatorics 2006 (San-Diego, 2006).
%H J.-C. Novelli and J.-Y. Thibon, <a href="http://journals.impan.gov.pl/fm/Inf/193-3-1.html">Hopf algebras and dendriform structures arising from parking functions</a>, Fundamenta Mathematicae 193 (2007), pp. 189-241; <a href="http://arxiv.org/abs/math/0511200">arXiv version</a>, 0511200 [math.CO], 2005.
%H 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 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 T. K. Petersen, <a href="http://www.springer.com/cda/content/document/cda_downloaddocument/9781493930906-c1.pdf">Chapter 2. Narayana numbers.</a> In: Eulerian Numbers. Birkhäuser Basel, 2015. doi:10.1007/978-1-4939-3091-3.
%H Vincent Pilaud and V. Pons, <a href="http://arxiv.org/abs/1606.09643">Permutrees</a>, arXiv:1606.09643 [math.CO], 2016-2017.
%H 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.
%H 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 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>
%H A. Sapounakis, I. Tasoulas and P. Tsikouras, <a href="http://dx.doi.org/10.1016/j.disc.2007.03.005">Counting strings in Dyck paths</a>, Discrete Math., 307 (2007), 2909-2924.
%H Paul R. F. Schumacher, <a href="https://www.emis.de/journals/JIS/VOL21/Schumacher/schu5.html">Descents in Parking Functions</a>, J. Int. Seq. 21 (2018), #18.2.3.
%H M. Sheppeard, <a href="http://vixra.org/pdf/1208.0242v6.pdf">Constructive motives and scattering</a> 2013 (p. 41). [_Tom Copeland_, Oct 03 2014]
%H R. P. Stanley, <a href="http://www-math.mit.edu/~rstan/pubs/pubfiles/12-2.pdf">Theory and application of plane partitions, II</a>. Studies in Appl. Math. 50 (1971), p. 259-279. <a href="http://doi.org/10.1002/sapm1971503259">DOI:10.1002/sapm1971503259</a>. Thm. 18.1.
%H R. A. Sulanke, <a href="http://math.boisestate.edu/~sulanke/recentpapindex.html">Moments, Narayana numbers and the cut and paste for lattice paths</a>
%H R. A. Sulanke, <a href="http://math.boisestate.edu/~sulanke/recentpapindex.html">Three-dimensional Narayana and Schröder numbers</a>
%H R. A. Sulanke, <a href="https://doi.org/10.1016/S0012-365X(97)00126-X">Catalan path statistics having the Narayana distribution</a>, Discrete Math., vol. 180 (1998), 369--389. [Gives additional contexts where Narayana numbers appear. - _N. J. A. Sloane_, Nov 11 2020]
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%H Tad White, <a href="https://arxiv.org/abs/2401.01462">Quota Trees</a>, arXiv:2401.01462 [math.CO], 2024. See pp. 19-20.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Noncrossing_partition">Noncrossing partition
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%H Anthony James Wood, <a href="https://hdl.handle.net/1842/36698">Nonequilibrium steady states from a random-walk perspective</a>, Ph. D. Thesis, The University of Edinburgh (Scotland, UK 2019), 44-46.
%H Anthony J. Wood, Richard A. Blythe, and Martin R. Evans, <a href="https://arxiv.org/abs/1908.00942">Combinatorial mappings of exclusion processes</a>, arXiv:1908.00942 [cond-mat.stat-mech], 2019.
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%H F. Yano and H. Yoshida, <a href="http://dx.doi.org/10.1016/j.disc.2007.03.050">Some set partition statistics in non-crossing partitions and generating functions</a>, Discr. Math., 307 (2007), 3147-3160.
%H 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.
%H 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.
%F a(n, k) = C(n-1, k-1)*C(n, k-1)/k for k!=0; a(n, 0)=0.
%F 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 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 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 T(n, k) = binomial(n-1, k-1)^2 - binomial(n-1, k)*binomial(n-1, k-2). - _David Callan_, Nov 02 2005
%F 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 Central column = A000891, (2n)!*(2n+1)! / (n!*(n+1)!)^2. - _Zerinvary Lajos_, Oct 29 2006
%F 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 From _Peter Bala_, Oct 22 2008: (Start)
%F Relation with Jacobi polynomials of parameter (1,1):
%F 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 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 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 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 G.f.: 1/(1-x-xy-x^2y/(1-x-xy-x^2y/(1-... (continued fraction). - _Paul Barry_, Sep 28 2010
%F E.g.f.: exp((1+y)x)*Bessel_I(1,2*sqrt(y)x)/(sqrt(y)*x). - _Paul Barry_, Sep 28 2010
%F 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 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 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 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 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 A166360(n-k) = T(n,k) mod 2. - _Reinhard Zumkeller_, Oct 10 2013
%F 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 Multiplying the n-th column by n! generates the revert of the unsigned Lah numbers, A089231. - _Tom Copeland_, Jan 07 2016
%F 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 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 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 T(n,k) = T(n,k-1)*C(n-k+2,2)/C(k,2). - _Yuchun Ji_, Dec 21 2020
%e The initial rows of the triangle are:
%e [1] 1
%e [2] 1, 1
%e [3] 1, 3, 1
%e [4] 1, 6, 6, 1
%e [5] 1, 10, 20, 10, 1
%e [6] 1, 15, 50, 50, 15, 1
%e [7] 1, 21, 105, 175, 105, 21, 1
%e [8] 1, 28, 196, 490, 490, 196, 28, 1
%e [9] 1, 36, 336, 1176, 1764, 1176, 336, 36, 1;
%e ...
%e For all n, 12...n (1 block) and 1|2|3|...|n (n blocks) are noncrossing set partitions.
%e Example of umbral representation:
%e A007318(5,k)=[1,5/1,5*4/(2*1),...,1]=(1,5,10,10,5,1),
%e so A001263(5,k)={1,b(5)/b(1),b(5)*b(4)/[b(2)*b(1)],...,1}
%e = [1,30/2,30*20/(6*2),...,1]=(1,15,50,50,15,1).
%e First = last term = b.(5!)/[b.(0!)*b.(5!)]= 1. - _Tom Copeland_, Sep 21 2011
%e 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 := (n,k)->binomial(n-1,k-1)*binomial(n,k-1)/k;
%p 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 # Alternatively, as a (0,0)-based triangle:
%p 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 T[n_, k_] := If[k==0, 0, Binomial[n-1, k-1] Binomial[n, k-1] / k];
%t Flatten[Table[Binomial[n-1,k-1] Binomial[n,k-1]/k,{n,15},{k,n}]] (* _Harvey P. Dale_, Feb 29 2012 *)
%t TRow[n_] := CoefficientList[Hypergeometric2F1[1 - n, -n, 2, x], x];
%t Table[TRow[n], {n, 1, 11}] // Flatten (* _Peter Luschny_, Mar 19 2018 *)
%t aot[n_]:=If[n==1,{{}},Join@@Table[Tuples[aot/@c],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
%t Table[Length[Select[aot[n],Length[Position[#,{}]]==k&]],{n,2,9},{k,1,n-1}] (* _Gus Wiseman_, Jan 23 2023 *)
%o (PARI) {a(n, k) = if(k==0, 0, binomial(n-1, k-1) * binomial(n, k-1) / k)};
%o (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 (Haskell)
%o a001263 n k = a001263_tabl !! (n-1) !! (k-1)
%o a001263_row n = a001263_tabl !! (n-1)
%o a001263_tabl = zipWith dt a007318_tabl (tail a007318_tabl) where
%o dt us vs = zipWith (-) (zipWith (*) us (tail vs))
%o (zipWith (*) (tail us ++ [0]) (init vs))
%o -- _Reinhard Zumkeller_, Oct 10 2013
%o (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 (Sage)
%o @CachedFunction
%o def T(n, k):
%o if k == n or k == 1: return 1
%o if k <= 0 or k > n: return 0
%o return binomial(n, 2) * (T(n-1, k)/((n-k)*(n-k+1)) + T(n-1, k-1)/(k*(k-1)))
%o for n in (1..9): print([T(n, k) for k in (1..n)]) # _Peter Luschny_, Oct 28 2014
%o (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 Other versions are in A090181 and A131198. - _Philippe Deléham_, Nov 18 2007
%Y Cf. variants: A181143, A181144. - _Paul D. Hanna_, Oct 13 2010
%Y Row sums give A000108 (Catalan numbers), n>0.
%Y Columns give A000217, A002415, A006542, A006857, A108679. A084938.
%Y Cf. A000372, A002083, A056932, A056939, A056940, A056941, A065329, A073345.
%Y A145596, A145597, A145598, A145599. - _Peter Bala_, Oct 22 2008
%Y 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 Cf. A016098 and A189232 for numbers of crossing set partitions.
%Y Cf. A243752.
%Y Cf. A089231, A103371, A135278.
%Y Cf. A002378, A007318, A010790, A089732, A105278, A119900, A132710, A134264.
%Y 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 Cf. A000081, A005043, A032027, A055277, A358371.
%K nonn,easy,tabl,nice,look,changed
%O 1,5
%A _N. J. A. Sloane_
%E Deleted certain dangerous or potentially dangerous links. - _N. J. A. Sloane_, Jan 30 2021
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