A062991 -id:A062991 - cp's OEIS frontend

A062992 Row sums of unsigned triangle A062991.

1, 3, 13, 67, 381, 2307, 14589, 95235, 636925, 4341763, 30056445, 210731011, 1493303293, 10678370307, 76957679613, 558403682307, 4075996839933, 29909606989827, 220510631755773, 1632599134961667, 12133359132082173

Offset: 0

Wolfdieter Lang, Jul 12 2001

a(n) = N(2; n,x=-1), with the polynomials N(2; n,x) defined in A062991.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Paul Barry, Characterizations of the Borel triangle and Borel polynomials, arXiv:2001.08799 [math.CO], 2020.
L. Guo and W. Y. Sit, Enumeration and generating functions of Rota-Baxter Words, Math. Comput. Sci. 4 (2010) 313-337.

Cf. A112707 (c(n, -m) triangle). Here m=2 is used. Row sums of A234950.

Cf. A000108, A062991, A064062, A119259.

a062992 = sum . a234950_row  -- Reinhard Zumkeller, Jan 12 2014

R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!( (1-2*x-Sqrt(1-8*x))/(2*x+2*x^2) )); // G. C. Greubel, Sep 27 2024

Table[2*Sum[(-1)^j*Binomial[2*n-2*j,n-j]/(n-j+1)*2^(n-j), {j,0,n}]-(-1)^n,{n,0,20}] (* Vaclav Kotesovec, Oct 13 2012 *)

a(n)=polcoeff((1-2*x-sqrt(1-8*x+x^2*O(x^n)))/(2*x+2*x^2),n)

a(n)=if(n<0,0,polcoeff(serreverse((x-x^2)/(1+x)^2+O(x^(n+2))),n+1)) \\ Ralf Stephan

def a(n): return hypergeometric([-n, n+1], [-n-1], 2)
[a(n).hypergeometric_simplify() for n in range(21)] # Peter Luschny, Nov 30 2014

a(n) = (-1)^(n+1) + 2*Sum_{j = 0..n} (-1)^j*C(n-j)*2^(n-j) with C(n) := A000108(n) (Catalan).
G.f.: A(x) = (2*c(2*x) - 1)/(1 + x) with c(x) the g.f. of A000108.
a(n) = (1/(n+1)) * Sum_{k = 0..n} binomial(2*n+2, n-k)*binomial(n+k, k). - Paul Barry, May 11 2005
Rewritten: a(n) = (1 - 2*c(n, -2))*(-1)^(n+1), n >= , with c(n, x) := Sum_{k = 0..n} C(k)*x^k and C(k) := A000108(k) (Catalan). - Wolfdieter Lang, Oct 31 2005
Recurrence: (n+1)*a(n) = (7*n-5)*a(n-1) + 4*(2*n-1)*a(n-2). - Vaclav Kotesovec, Oct 13 2012
a(n) ~ 2^(3*n+4)/(9*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 13 2012
a(n) = hypergeometric([-n, n+1], [-n-1], 2). - Peter Luschny, Nov 30 2014
G.f.: A(x) = exp( Sum_{n >= 1} A119259(n)*x^n/n ). - Peter Bala, Jun 08 2023

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).

Original entry on oeis.org

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, 1, 7, 27, 75, 165, 297, 429, 429, 1, 8, 35, 110, 275, 572, 1001, 1430, 1430, 1, 9, 44, 154, 429, 1001, 2002, 3432, 4862, 4862, 1, 10, 54, 208, 637, 1638, 3640, 7072, 11934

Offset: 0

Wouter Meeussen

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

			Triangle begins in row n=0 with 0 <= k <= n:
  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;
  1, 7, 27,  75, 165,  297,  429,  429;
  1, 8, 35, 110, 275,  572, 1001, 1430, 1430;
  1, 9, 44, 154, 429, 1001, 2002, 3432, 4862, 4862;

William Feller, Introduction to Probability Theory and its Applications, vol. I, ed. 2, chap. 3, sect. 1,2.
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. E. Knuth, TAOCP, Vol. 4, Section 7.2.1.6, Eq. 22, p. 451.
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.
M. Bellon, Query 5467, L'Intermédiaire des Mathématiciens, 4 (1925), 11; H. Ory, 4 (1925), 120. - N. J. A. Sloane, Mar 09 2022
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).
M. Welsch, Note #371, L'Intermédiaire des Mathématiciens, 2 (1895), pp. 235-237. - N. J. A. Sloane, Mar 02 2022

T. D. Noe, Rows n = 0..100 of triangle, flattened
Erik Aas, Arvind Ayyer, Svante Linusson and Samu Potka, The exact phase diagram for a semipermeable TASEP with nonlocal boundary jumps, arXiv:1902.02019 [cond-mat.stat-mech], 2019.
Ron M. Adin, E. Bagno, and Y. Roichman, Block decomposition of permutations and Schur-positivity, arXiv:1611.06979 [math.CO], 2016-2017.
Kassie Archer, Abigail Bishop, Alexander Diaz-Lopez, Luis David Garcia Puente, Darren Glass, and Joel Louwsma, Arithmetical structures on bidents, arXiv:1903.01393 [math.CO], 2019.
J. L. Arregui, Tangent and Bernoulli numbers related to Motzkin and Catalan numbers by means of numerical triangles, arXiv:math/0109108 [math.NT], 2001.
Jean-Christophe Aval, Multivariate Fuss-Catalan numbers, arXiv:0711.0906 [math.CO], 2007.
Jean-Christophe Aval, Multivariate Fuss-Catalan numbers, Discrete Math., 308 (2008), 4660-4669.
Axel Bacher, Antonio Bernini, Luca Ferrari, Benjamin Gunby, Renzo Pinzani, and Julian West, The Dyck pattern poset Discrete Math. 321 (2014), 12--23. MR3154009.
D. F. Bailey, Counting arrangements of 1's and-1's, Mathematics Magazine, 69 (1996): 128-131. See table on p. 129.
Elena Barcucci, Alberto Del Lungo, Elisa Pergola, and Renzo Pinzani, A methodology for plane tree enumeration, 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).
E. Barcucci and M. C. Verri, Some more properties of Catalan numbers, Discrete Math., 102 (1992), 229-237.
J.-L. Baril, C. Khalil, and V. Vajnovszki, Catalan and Schröder permutations sortable by two restricted stacks, arXiv:2004.01812 [cs.DM], 2020.
Paul Barry, Riordan Arrays, Orthogonal Polynomials as Moments, and Hankel Transforms, J. Int. Seq. 14 (2011) # 11.2.2, example 3.
Paul Barry, On the Inverses of a Family of Pascal-Like Matrices Defined by Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.5.6.
Paul Barry, Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.
Paul Barry, Characterizations of the Borel triangle and Borel polynomials, arXiv:2001.08799 [math.CO], 2020.
Paul Barry, On the inversion of Riordan arrays, arXiv:2101.06713 [math.CO], 2021.
Paul Barry and A. Hennessy, Meixner-Type Results for Riordan Arrays and Associated Integer Sequences, J. Int. Seq. 13 (2010) # 10.9.4, example 3.
F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999) 73-112.
A. Bernini, L. Ferrari, R. Pinzani, and J. West, The Dyck pattern poset, arXiv:1303.3785 [math.CO], 2013.
N. Borie, Combinatorics of simple marked mesh patterns in 132-avoiding permutations, arXiv:1311.6292 [math.CO], 2013.
M. Bousquet-Mélou and M. Petkovsek, Linear recurrences with constant coefficients: the multivariate case, Discrete Math. 225 (2000), 51-75.
Jasper Bown, Peter Kagey, Alan Kappler, Michael E. Orrison, and Jayden Thadani, Preference-restricted parking functions, arXiv:2507.11701 [math.CO], 2025. See p. 11.
Benjamin Braun, Hugo Corrales, Scott Corry, Luis David García Puente, Darren Glass, Nathan Kaplan, Jeremy L. Martin, Gregg Musiker, and Carlos E. Valencia, Counting Arithmetical Structures on Paths and Cycles. arXiv:1701.06377 [math.CO], 2017.
E. H. M. Brietzke, An identity of Andrews and a new method for the Riordan array proof of combinatorial identities, Discrete Math., 308 (2008), 4246-4262.
S. Brlek, E. Duchi, E. Pergola, and S. Rinaldi, On the equivalence problem for succession rules, Discr. Math., 298 (2005), 142-154.
Hong Duc Bui, Counting Number of Triangulations of Point Sets: Reinterpreting and Generalizing the Triangulation Polynomials, arXiv:2504.09615 [cs.CG], 2025. See p. 4.
Steve Butler, R. Graham, and C. H. Yan, Parking distributions on trees, European Journal of Combinatorics 65 (2017), 168-185.
R. J. Cano, Catalan's books
L. Carlitz, Sequences, paths, ballot numbers
Douglas M. Chen, On the Structure of Permutation Invariant Parking, arXiv:2311.15699 [math.CO], 2023. See p. 16.
Lapo Cioni and Luca Ferrari, Preimages under the Queuesort algorithm, arXiv preprint arXiv:2102.07628 [math.CO], 2021; Discrete Math., 344 (2021), #112561.
Ari Cruz, Pamela E. Harris, Kimberly J. Harry, Jan Kretschmann, Matt McClinton, Alex Moon, John O. Museus, and Eric Redmon, On some discrete statistics of parking functions, arXiv:2312.16786 [math.CO], 2023.
Italo J. Dejter, A new approach to the middle levels via a Catalan-number system of numeration, 2015.
Italo J. Dejter, A numeral system for the middle levels, preprint, 2014. [See Section 2. - _N. J. A. Sloane_, Apr 06 2014]
Italo J. Dejter, Dihedral-symmetry middle-levels problem via a Catalan system of numeration, preprint, 2015.
Italo J. Dejter, The role of restricted growth strings in the two middle levels of the Boolean lattice B_(2k+1), University of Puerto Rico, 2018.
Italo J. Dejter, Reinterpreting Mütze's Theorem via Natural Enumeration of Ordered Rooted Trees, arXiv:1911.02100 [math.CO], 2019.
Italo J. Dejter, A numeral system for the middle-levels graphs, Elec. J. Graph Theory and Applications (2021) Vol. 9, No. 1, 137-156. See p. 139.
B. Derrida, E. Domany, and D. Mukamel, An exact solution of a one-dimensional asymmetric exclusion model with open boundaries, 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).
E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.
Filippo Disanto, Some Statistics on the Hypercubes of Catalan Permutations, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.2.
Paul Drube, Generating Functions for Inverted Semistandard Young Tableaux and Generalized Ballot Numbers, arXiv:1606.04869 [math.CO], 2016.
Richard Ehrenborg, Gábor Hetyei, and Margaret Readdy, Classification of uniform flag triangulations of the Legendre polytope, arXiv:1901.07113 [math.CO], 2019.
R. Ehrenborg, S. Kitaev, and E. Steingrimsson, Number of cycles in the graph of 312-avoiding permutations, arXiv:1310.1520 [math.CO], 2013.
W. J. R. Eplett, A note about the Catalan triangle, Discrete Math. 25(1979), no. 3, 289--291. MR0534947 (80i:05007)
Jackson Evoniuk, Steven Klee, and Van Magnan, Enumerating Minimal Length Lattice Paths, J. Int. Seq., Vol. 21 (2018), Article 18.3.6.
G. Feinberg and K.-H. Lee, Homogeneous representations of KLR-algebras and fully commutative elements, arXiv:1401.0845 [math.RT], 2014.
I. Fanti, A. Frosini, E. Grazzini, R. Pinzani, and S. Rinaldi, Characterization and enumeration of some classes of permutominoes, PU. M. A., Vol. 18 (2007), No. 3-4, pp. 265-290.
Dominique Foata and Guo-Niu Han, The doubloon polynomial triangle, Ram. J. 23 (2010), 107-126.
Dominique Foata and Guo-Niu Han, Doubloons and new q-tangent numbers, Quart. J. Math. 62 (2) (2011) 417-432.
H. G. Forder, Some problems in combinatorics, Math. Gazette, vol. 45, 1961, 199-201.
C. A. Francisco, J. Mermin, and J. Schweig, Catalan numbers, binary trees, and pointed pseudotriangulations, preprint 2013; European Journal of Combinatorics, Volume 45, April 2015, Pages 85-96.
Shishuo Fu, Zhicong Lin, and Yaling Wang, A combinatorial bijection on di-sk trees, arXiv:2011.11302 [math.CO], 2020.
Ling Gao, Graph assembly for spider and tadpole graphs, Master's Thesis, Cal. State Poly. Univ. (2023). See p. 12.
Ira Gessel, Super ballot numbers.
Samuele Giraudo, Tree series and pattern avoidance in syntax trees, arXiv:1903.00677 [math.CO], 2019.
Claude Godrèche and Jean-Marc Luck, On the first positive position of a random walker, arXiv:2501.17268 [cond-mat.stat-mech], 2025.
Niket Gowravaram, A Variation of the nil-Temperley-Lieb algebras of type A, Preprint, 2015.
J. M. Hammersley, An undergraduate exercise in manipulation, Math. Scientist, 14 (1989), 1-23.
J. M. Hammersley, An undergraduate exercise in manipulation, Math. Scientist, 14 (1989), 1-23. (Annotated scanned copy)
Pamela E. Harris, Thiago Holleben, J. Carlos Martínez Mori, Amanda Priestley, Keith Sullivan, and Per Wagenius, A Probabilistic Parking Process and Labeled IDLA, arXiv:2501.11718 [math.PR], 2025. See pp. 8, 28.
David He and Daniel Tubbenhauer, Tensor powers of representations of (diagram) monoids, arXiv:2508.04054 [math.RT], 2025. See p. 7.
Nickolas Hein and Jia Huang, Variations of the Catalan numbers from some nonassociative binary operations, arXiv:1807.04623 [math.CO], 2018.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
F. Hivert, J.-C. Novelli, and J.-Y. Thibon, The Algebra of Binary Search Trees, Theoretical Computer Science, 339 (2005), 129-165.
R. L. Hudson and Y. Pei, On a causal quantum stochastic double product integral related to Lévy area, Research Gate, 2015.
R. L. Hudson and Y. Pei, On a quantum causal stochastic double product integral related to Levy area, arXiv:1506.04294 [math-ph], 2015.
Brant Jones, Avoiding patterns and making the best choice, arXiv:1812.00963 [math.CO], 2018.
Cemil Karaçam and Alper Vural, Enumerating 2D and 3D lattice paths with arbitrary steps, Mathematica Bohemica, pp. 1-13 (2025). See p. 12.
Antii Karttunen, Some notes on Catalan's Triangle.
Lord C. Kavi and Michael W. Newman, Counting closed walks in infinite regular trees using Catalan and Borel's triangles, arXiv:2212.08795 [math.CO], 2022.
Dongsu Kim and Zhicong Lin, Refined restricted inversion sequences, arXiv:1706.07208 [math.CO], 2017.
W. Krandick, Trees and jumps and real roots, J. Computational and Applied Math., 162, 2004, 51-55.
G. Kreweras, Sur les éventails de segments, Cahiers du Bureau Universitaire de Recherche Opérationelle, Cahier no. 15, Paris, 1970, pp. 3-41.
G. Kreweras, Sur les éventails de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #15 (1970), 3-41. [Annotated scanned copy]
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. [Annotated scanned copy]
A. Laradji and A. Umar, On certain finite semigroups of order-decreasing transformations I, Semigroup Forum 69 (2004), 184-200.
Jeong-Yup Lee, Dong-il Lee, and Sungsoon Kim, Gröbner-Shirshov bases for Temperley-Lieb algebras of the complex reflection group of type G(d,1,n), arXiv:1808.06523 [math.RA], 2018.
Kyu-Hwan Lee and Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.
Shao-Hua Liu, The operators F_i on permutations, 132-avoiding permutations and inversions, Discrete Math., 342 (2019), 2402-2414.
Jiaxi Lu and Yuanzhe Ding, A skeleton model to enumerate standard puzzle sequences, arXiv:2106.09471 [math.CO], 2021.
D. Merlini et al., Underdiagonal lattice paths with unrestricted steps, Discrete Appl. Math., 91 (1999), 197-213.
D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344 (Table I).
Andrew Misseldine, Counting Schur Rings over Cyclic Groups, arXiv:1508.03757 [math.RA], 2015.
J.-C. Novelli and J.-Y. Thibon, Noncommutative Symmetric Functions and Lagrange Inversion, arXiv:math/0512570 [math.CO], 2005-2006.
M. A. A. Obaid, S. K. Nauman, W. M. Fakieh, and C. M. Ringel, The numbers of support-tilting modules for a Dynkin algebra, 2014 and J. Int. Seq. 18 (2015) 15.10.6.
P. Pagacz and M. Wojtylak, On the spectral properties of a class of H-selfadjoint random matrices and the underlying combinatorics, arXiv:1310.2122 [math.PR], 2013.
R. Parviainen, Permutations, cycles and the pattern 2-13, arXiv:math/0607793 [math.CO], 2006.
R. Pemantle and M. C. Wilson, Twenty Combinatorial Examples of Asymptotics Derived from Multivariate Generating Functions, SIAM Rev., 50 (2008), no. 2, 199-272. See p. 261
Planet Math, Lattice Paths and ballot numbers.
L. Pudwell, Avoiding an Ordered Partition of Length 3, 2013.
C. M. Ringel, The Catalan combinatorics of the hereditary artin algebras, arXiv:1502.06553 [math.RT], 2015.
Gabriel Bravo Rios and Agustin Moreno Cañadas, Dyck Paths in Representation Theory of Algebras, National University of Colombia (2020).
A. Robertson, D. Saracino, and D. Zeilberger, Refined restricted permutations, arXiv:math/0203033 [math.CO], 2002.
L. W. Shapiro, A Catalan triangle, Discrete Math., 14, 83-90, 1976.
T. Sillke, Catalan's numbers
Hermann Stamm-Wilbrandt, Visualization for all well-formed strings of 4 parenthesis pairs and relation to T(n,k).
R. A. Sulanke, Guessing, ballot numbers and refining Pascal's triangle
Yidong Sun, A simple bijection between binary trees and colored ternary trees, El. J. Combinat. 17 (2010) #N20
Paweł J. Szabłowski, Yet another way of calculating moments of the Kesten's distribution and its consequences, arXiv:2106.10461 [math.CO], 2021.
Benjamin Testart, Completing the enumeration of inversion sequences avoiding one or two patterns of length 3, arXiv:2407.07701 [math.CO], 2024. See pp. 9, 29.
Rafael Vazquez and M. Krstic, Boundary control of a singular reaction-diffusion equation on a disk, arXiv:1601.02010 [math.OC], 2016.
Luis Verde-Star, A Matrix Approach to Generalized Delannoy and Schröder Arrays, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.
Eric Weisstein's World of Mathematics, Catalan's Triangle
Eric Weisstein's World of Mathematics, Nonnegative Partial Sum
Tad White, Quota Trees, arXiv:2401.01462 [math.CO], 2024. See p. 17.
Martha Yip, Rook Placements and Jordan Forms of Upper-Triangular Nilpotent Matrices, arXiv:1703.00057 [math.CO], 2017; The Electronic Journal of Combinatorics 25(1) (2018), #P1.68.

The following are all versions of (essentially) the same Catalan triangle: A009766, A008315, A028364, A030237, A047072, A053121, A059365, A062103, A099039, A106566, A130020, A140344.
Cf. A062745, A214292.
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, ...
Row sums as well as the sums of the squares of the shallow diagonals give Catalan numbers (A000108).
Reflected version of A033184.
Diagonals give A000108, A000108, A000245, A002057, A000344, A003517, A000588, A003518, A003519, A001392, ...
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).

Flat(List([0..10],n->List([0..n],m->Binomial(n+m,n)*(n-m+1)/(n+1)))); # Muniru A Asiru, Feb 18 2018

a009766 n k = a009766_tabl !! n !! k
a009766_row n = a009766_tabl !! n
a009766_tabl = iterate (\row -> scanl1 (+) (row ++ [0])) [1]
-- Reinhard Zumkeller, Jul 12 2012

[[Binomial(n+k,n)*(n-k+1)/(n+1): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Mar 07 2019

A009766 := proc(n,k) binomial(n+k,n)*(n-k+1)/(n+1); end proc:
seq(seq(A009766(n,k), k=0..n), n=0..10); # R. J. Mathar, Dec 03 2010

Flatten[NestList[Append[Accumulate[#], Last[Accumulate[#]]] &, {1}, 9]] (* Birkas Gyorgy, May 19 2012 *)
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 *)

{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 */

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

from math import comb, isqrt
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

@CachedFunction
def ballot(p,q):
    if p == 0 and q == 0: return 1
    if p < 0 or p > q: return 0
    S = ballot(p-2, q) + ballot(p, q-2)
    if q % 2 == 1: S += ballot(p-1, q-1)
    return S
A009766 = lambda n, k: ballot(2*k, 2*n)
for n in (0..7): [A009766(n, k) for k in (0..n)]  # Peter Luschny, Mar 05 2014

[[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

T(n, m) = binomial(n+m, n)*(n-m+1)/(n+1), 0 <= m <= n.
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.
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
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
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
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
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
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
T(n,k) = binomial(n + k + 1, k) - 2*binomial(n + k, k - 1), for 0 <= k <= n. - David Callan, Jun 15 2022

A088617 Triangle read by rows: T(n,k) = C(n+k,n)*C(n,k)/(k+1), for n >= 0, k = 0..n.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 6, 10, 5, 1, 10, 30, 35, 14, 1, 15, 70, 140, 126, 42, 1, 21, 140, 420, 630, 462, 132, 1, 28, 252, 1050, 2310, 2772, 1716, 429, 1, 36, 420, 2310, 6930, 12012, 12012, 6435, 1430, 1, 45, 660, 4620, 18018, 42042, 60060, 51480, 24310, 4862

Offset: 0

N. J. A. Sloane, Nov 23 2003

Row sums: A006318 (Schroeder numbers). Essentially same as triangle A060693 transposed.
T(n,k) is number of Schroeder paths (i.e., consisting of steps U=(1,1), D=(1,-1), H=(2,0) and never going below the x-axis) from (0,0) to (2n,0), having k U's. E.g., T(2,1)=3 because we have UHD, UDH and HUD. - Emeric Deutsch, Dec 06 2003
Little Schroeder numbers A001003 have a(n) = Sum_{k=0..n} A088617(n,k)*(-1)^(n-k)*2^k. - Paul Barry, May 24 2005
Conjecture: The expected number of U's in a Schroeder n-path is asymptotically Sqrt[1/2]*n for large n. - David Callan, Jul 25 2008
T(n, k) is also the number of order-preserving and order-decreasing partial transformations (of an n-chain) of width k (width(alpha) = |Dom(alpha)|). - Abdullahi Umar, Oct 02 2008
The antidiagonals of this lower triangular matrix are the rows of A055151. - Tom Copeland, Jun 17 2015

			Triangle begins:
  [0] 1;
  [1] 1,  1;
  [2] 1,  3,   2;
  [3] 1,  6,  10,    5;
  [4] 1, 10,  30,   35,    14;
  [5] 1, 15,  70,  140,   126,    42;
  [6] 1, 21, 140,  420,   630,   462,   132;
  [7] 1, 28, 252, 1050,  2310,  2772,  1716,   429;
  [8] 1, 36, 420, 2310,  6930, 12012, 12012,  6435,  1430;
  [9] 1, 45, 660, 4620, 18018, 42042, 60060, 51480, 24310, 4862;

Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 449.

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150)
Anwar Al Ghabra, K. Gopala Krishna, Patrick Labelle, and Vasilisa Shramchenko, Enumeration of multi-rooted plane trees, arXiv:2301.09765 [math.CO], 2023.
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
Paul Barry, Continued fractions and transformations of integer sequences, JIS 12 (2009) 09.7.6.
Paul Barry, Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.
Paul Barry, On the inversion of Riordan arrays, arXiv:2101.06713 [math.CO], 2021.
Manosij Ghosh Dastidar and Michael Wallner, Bijections and congruences involving lattice paths and integer compositions, arXiv:2402.17849 [math.CO], 2024. See p. 16.
Samuele Giraudo, Tree series and pattern avoidance in syntax trees, arXiv:1903.00677 [math.CO], 2019.
Hsien-Kuei Hwang and Satoshi Kuriki, Integrated empirical measures and generalizations of classical goodness-of-fit statistics, arXiv:2404.06040 [math.ST], 2024. See p. 11.
C. Jordan, Calculus of Finite Differences, Budapest, 1939. [Annotated scans of pages 448-450 only]
A. Kirillov, On Some Quadratic Algebras I 1/2: Combinatorics of Dunkl and Gaudin Elements, Schubert, Grothendieck, Fuss-Catalan, Universal Tutte and Reduced Polynomials, arXiv preprint arXiv:1502.00426 [math.RT], 2016.
M. Klazar, On numbers of Davenport-Schinzel sequences, Discr. Math., 185 (1998), 77-87.
Paul W. Lapey and Aaron Williams, A Shift Gray Code for Fixed-Content Łukasiewicz Words, Williams College, 2022.
A. Laradji and A. Umar, A. Combinatorial results for semigroups of order-preserving partial transformations, Journal of Algebra 278, (2004), 342-359.
A. Laradji and A. Umar, Combinatorial results for semigroups of order-decreasing partial transformations, J. Integer Seq. 7 (2004), 04.3.8.
Jason P. Smith, The poset of graphs ordered by induced containment, arXiv:1806.01821 [math.CO], 2018.

Diagonals give A000217, A034827, A000910, A088625, A088626, A000108, A001700, A002457, A002802, A002803.
Cf. A000108, A001003, A006318, A060693, A084938, A085478.
Cf. A033282, A055151, A123160.
Cf. A001263, A011973, A055151, A060693, A074909, A086810, A107131.

[[Binomial(n+k,n)*Binomial(n,k)/(k+1): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Jun 18 2015

R := n -> simplify(hypergeom([-n, n + 1], [2], -x)):
Trow := n -> seq(coeff(R(n, x), x, k), k = 0..n):
seq(print(Trow(n)), n = 0..9); # Peter Luschny, Apr 26 2022

Table[Binomial[n+k, n] Binomial[n, k]/(k+1), {n,0,10}, {k,0,n}]//Flatten (* Michael De Vlieger, Aug 10 2017 *)

{T(n, k)= if(k+1, binomial(n+k, n)*binomial(n, k)/(k+1))}

flatten([[binomial(n+k, 2*k)*catalan_number(k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 22 2022

Triangle T(n, k) read by rows; given by [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...] DELTA [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...] where DELTA is Deléham's operator defined in A084938.
T(n, k) = A085478(n, k)*A000108(k); A000108 = Catalan numbers. - Philippe Deléham, Dec 05 2003
Sum_{k=0..n} T(n, k)*x^k*(1-x)^(n-k) = A000108(n), A001003(n), A007564(n), A059231(n), A078009(n), A078018(n), A081178(n), A082147(n), A082181(n), A082148(n), A082173(n) for x = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. - Philippe Deléham, Aug 18 2005
Sum_{k=0..n} T(n,k)*x^k = (-1)^n*A107841(n), A080243(n), A000007(n), A000012(n), A006318(n), A103210(n), A103211(n), A133305(n), A133306(n), A133307(n), A133308(n), A133309(n) for x = -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8 respectively. - Philippe Deléham, Oct 18 2007
O.g.f. (with initial 1 excluded) is the series reversion with respect to x of (1-t*x)*x/(1+x). Cf. A062991 and A089434. - Peter Bala, Jul 31 2012
G.f.: 1 + (1 - x - T(0))/y, where T(k) = 1 - x*(1+y)/( 1 - x*y/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 03 2013
From Peter Bala, Jul 20 2015: (Start)
O.g.f. A(x,t) = ( 1 - x - sqrt((1 - x)^2 - 4*x*t) )/(2*x*t) = 1 + (1 + t)*x + (1 + 3*t + 2*t^2)*x^2 + ....
1 + x*(dA(x,t)/dx)/A(x,t) = 1 + (1 + t)*x + (1 + 4*t + 3*t^2)*x^2 + ... is the o.g.f. for A123160.
For n >= 1, the n-th row polynomial equals (1 + t)/(n+1)*Jacobi_P(n-1,1,1,2*t+1). Removing a factor of 1 + t from the row polynomials gives the row polynomials of A033282. (End)
From Tom Copeland, Jan 22 2016: (Start)
The o.g.f. G(x,t) = {1 - (2t+1) x - sqrt[1 - (2t+1) 2x + x^2]}/2x = (t + t^2) x + (t + 3t^2 + 2t^3) x^2 + (t + 6t^2 + 10t^3 + 5t^3) x^3 + ... generating shifted rows of this entry, excluding the first, was given in my 2008 formulas for A033282 with an o.g.f. f1(x,t) = G(x,t)/(1+t) for A033282. Simple transformations presented there of f1(x,t) are related to A060693 and A001263, the Narayana numbers. See also A086810.
The inverse of G(x,t) is essentially given in A033282 by x1, the inverse of f1(x,t): Ginv(x,t) = x [1/(t+x) - 1/(1+t+x)] = [((1+t) - t) / (t(1+t))] x - [((1+t)^2 - t^2) / (t(1+t))^2] x^2 + [((1+t)^3 - t^3) / (t(1+t))^3] x^3 - ... . The coefficients in t of Ginv(xt,t) are the o.g.f.s of the diagonals of the Pascal triangle A007318 with signed rows and an extra initial column of ones. The numerators give the row o.g.f.s of signed A074909.
Rows of A088617 are shifted columns of A107131, whose reversed rows are the Motzkin polynomials of A055151, related to A011973. The diagonals of A055151 give the rows of A088671, and the antidiagonals (top to bottom) of A088617 give the rows of A107131 and reversed rows of A055151. The diagonals of A107131 give the columns of A055151. The antidiagonals of A088617 (bottom to top) give the rows of A055151.
(End)
T(n, k) = [x^k] hypergeom([-n, 1 + n], [2], -x). - Peter Luschny, Apr 26 2022

A005587 a(n) = n(n+5)(n+6)*(n+7)/24.

Original entry on oeis.org

0, 14, 42, 90, 165, 275, 429, 637, 910, 1260, 1700, 2244, 2907, 3705, 4655, 5775, 7084, 8602, 10350, 12350, 14625, 17199, 20097, 23345, 26970, 31000, 35464, 40392, 45815, 51765, 58275, 65379, 73112, 81510, 90610, 100450, 111069, 122507, 134805, 148005

Offset: 0

N. J. A. Sloane

a(n) = number of Standard Young Tableaux of shape (n+3,4). - David Callan, Aug 17 2004
a(n) = A214292(n+6,3). - Reinhard Zumkeller, Jul 12 2012
a(n) for n > 0 is the number of n-extended coalescent histories for a matching caterpillar gene tree and species tree with 5 leaves. - Noah A Rosenberg, Jun 16 2022

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 796.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Richard K. Guy, Letter to N. J. A. Sloane, Feb 1988.
Richard K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
N. A. Rosenberg, Counting coalescent histories, J. Comput. Biol. 14 (2007), 360-377.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Fifth diagonal of Catalan triangle A033184. Fifth column of Catalan triangle A009766.

Numerator polynomial 14 - 28x + 20x^2 - 5x^3 from fourth row of triangle A062991.

[n*(n+5)*(n+6)*(n+7)/24: n in [0..40]]; // Vincenzo Librandi, Mar 20 2013

A005587:=z*(-14+28*z-20*z**2+5*z**3)/(z-1)**5; # Simon Plouffe in his 1992 dissertation
seq(numbperm(n,4)/24-numbperm(n,3)/6, n=7..46); # Zerinvary Lajos, May 20 2008
a:=n->(sum(numbcomp(n,4), j=9..n)):seq(a(n)/4, n=8..47); # Zerinvary Lajos, Aug 26 2008

Table[n (n + 5) (n + 6) (n + 7)/24, {n, 0, 60}] (* Vladimir Joseph Stephan Orlovsky, Jun 22 2011 *)
LinearRecurrence[{5,-10,10,-5,1},{0,14,42,90,165},40] (* Harvey P. Dale, Aug 17 2017 *)

x='x+O('x^50); concat([0], Vec((14 - 28*x + 20*x^2 - 5*x^3) / (1 - x)^5)) \\ G. C. Greubel, Jul 01 2017

G.f.: (14 - 28*x + 20*x^2 - 5*x^3) / (1 - x)^5.
a(n) = C(7+n, 4) - C(7+n, 3). - Zerinvary Lajos, Dec 09 2005
E.g.f.: (1/24)*x*(336 + 168*x + 24*x^2 + x^3)*exp(x). - G. C. Greubel, Jul 01 2017
From Amiram Eldar, Jun 28 2022: (Start)
Sum_{n>=1} 1/a(n) = 153/1225.
Sum_{n>=1} (-1)^(n+1)/a(n) = 288*log(2)/35 - 20759/3675. (End)
a(n) = A024191(n+1)-5. - R. J. Mathar, Nov 22 2024

M4929 (this sequence) and M4930 were the same.
More terms from Matthew Conroy, Jan 16 2006
Plouffe Maple line edited by N. J. A. Sloane, May 13 2008

A064059 Seventh column of Catalan triangle A009766.

Original entry on oeis.org

132, 429, 1001, 2002, 3640, 6188, 9996, 15504, 23256, 33915, 48279, 67298, 92092, 123970, 164450, 215280, 278460, 356265, 451269, 566370, 704816, 870232, 1066648, 1298528, 1570800, 1888887, 2258739, 2686866, 3180372, 3746990, 4395118, 5133856, 5973044

Offset: 0

Wolfdieter Lang, Sep 13 2001

G. C. Greubel, Table of n, a(n) for n = 0..1000
Richard K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article 00.1.6.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A000096, A005586, A005587, A005557 (third to sixth column).

A064059:= func< n | (n+1)*Binomial(n+12,5)/6 >;
[A064059(n): n in [0..40]]; // G. C. Greubel, Sep 27 2024

[seq(binomial(n+1,6)-2*binomial(n,5),n=12..55)]; # Zerinvary Lajos, Jul 19 2006

CoefficientList[Series[(42 z^5-252 z^4+616 z^3-770 z^2+495 z-132)/(z-1)^7, {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 22 2011 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{132,429,1001,2002,3640,6188,9996},40] (* Harvey P. Dale, Jan 08 2025 *)

def A064059(n): return (n+1)*binomial(n+12,5)//6
[A064059(n) for n in range(41)] # G. C. Greubel, Sep 27 2024

G.f.: (132-495*x+770*x^2-616*x^3+252*x^4-42*x^5)/(1-x)^7; numerator polynomial is N(2;5, x) from A062991.
a(n) = A009766(n+6, 6) = (n+1)*binomial(n+12,5)/6.
a(n) = binomial(n+13,6) - 2*binomial(n+12,5). - Zerinvary Lajos, Jul 19 2006
a(n) = A214292(n+11,5). - Reinhard Zumkeller, Jul 12 2012
From Amiram Eldar, Sep 06 2022: (Start)
Sum_{n>=0} 1/a(n) = 25961/2134440.
Sum_{n>=0} (-1)^n/a(n) = 4160*log(2)/77 - 79917773/2134440. (End)

A005557 Number of walks on square lattice.

Original entry on oeis.org

42, 132, 297, 572, 1001, 1638, 2548, 3808, 5508, 7752, 10659, 14364, 19019, 24794, 31878, 40480, 50830, 63180, 77805, 95004, 115101, 138446, 165416, 196416, 231880, 272272, 318087, 369852, 428127, 493506, 566618, 648128, 738738, 839188, 950257, 1072764

Offset: 0

N. J. A. Sloane

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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Richard K. Guy, Letter to N. J. A. Sloane, May 1990.
Richard K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article 00.1.6.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Sixth diagonal of Catalan triangle A033184.
Sixth column of Catalan triangle A009766.
Cf. A062991, A214292.

List([0..30],n->(n+1)*Binomial(n+10,4)/5); # Muniru A Asiru, Apr 10 2018

[(n+1)*Binomial(n+10, 4)/5: n in [0..40]]; // Vincenzo Librandi, Mar 20 2013

[seq(binomial(n,5)-binomial(n,3),n=9..55)]; # Zerinvary Lajos, Jul 19 2006
A005557:=(42-120*z+135*z**2-70*z**3+14*z**4)#(z-1)**6; # conjectured by Simon Plouffe in his 1992 dissertation

CoefficientList[Series[(14 z^4 - 70 z^3 + 135 z^2 - 120 z + 42)/(z - 1)^6, {z, 0, 200}], z] (* Vladimir Joseph Stephan Orlovsky, Jun 22 2011 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{42,132,297,572,1001,1638},40] (* Harvey P. Dale, Feb 22 2024 *)

a(n)=(n+1)*binomial(n+10,4)/5 \\ Charles R Greathouse IV, Oct 21 2022

a(n) = A009766(n+5, 5) = (n+1)*binomial(n+10, 4)/5.
G.f.: (42 - 120*x + 135*x^2 - 70*x^3 + 14*x^4)/(1-x)^6; numerator polynomial is N(2;4, x) from A062991.
a(n) = binomial(n+9,5) - binomial(n+9,3). - Zerinvary Lajos, Jul 19 2006
a(n) = A214292(n+9, 4). - Reinhard Zumkeller, Jul 12 2012
From Amiram Eldar, Sep 06 2022: (Start)
Sum_{n>=0} 1/a(n) = 2509/63504.
Sum_{n>=0} (-1)^n/a(n) = 951395/63504 - 1360*log(2)/63. (End)

More terms and formula from Wolfdieter Lang, Sep 04 2001

A064061 Eighth column of Catalan triangle A009766.

Original entry on oeis.org

429, 1430, 3432, 7072, 13260, 23256, 38760, 62016, 95931, 144210, 211508, 303600, 427570, 592020, 807300, 1085760, 1442025, 1893294, 2459664, 3164480, 4034712, 5101360, 6399888, 7970688, 9859575, 12118314, 14805180, 17985552, 21732542, 26127660, 31261516

Offset: 0

Wolfdieter Lang, Sep 13 2001

G. C. Greubel, Table of n, a(n) for n = 0..1000
Richard K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article 00.1.6.
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Cf. A009766, A064059 (seventh column), A062991, A214292.

A064061:= func< n | (n+1)*Binomial(n+14, 6)/7 >;
[A064061(n): n in [0..40]]; // G. C. Greubel, Sep 28 2024

[seq(binomial(n,7)-binomial(n,5),n=13..37)]; # Zerinvary Lajos, Nov 25 2006

CoefficientList[Series[(132*z^6 - 924*z^5 + 2730*z^4 - 4368*z^3 + 4004*z^2 - 2002*z + 429)/(z - 1)^8, {z, 0, 100}], z] (* Vladimir Joseph Stephan Orlovsky, Jul 16 2011 *)
Table[Binomial[n,7]-Binomial[n,5],{n,13,50}] (* or *) LinearRecurrence[ {8,-28,56,-70,56,-28,8,-1},{429,1430,3432,7072,13260,23256,38760,62016},40] (* Harvey P. Dale, Sep 03 2015 *)

def A064061(n): return (n+1)*binomial(n+14,6)//7
[A064061(n) for n in range(41)] # G. C. Greubel, Sep 28 2024

a(n) = A009766(n+7, 7) = (n+1)*binomial(n+14, 6)/7.
G.f.: (429-2002*x+4004*x^2-4368*x^3+2730* x^4-924*x^5+132*x^6)/(1-x)^8; numerator polynomial is N(2;6, x) from A062991.
a(n) = C(n+13,7) - C(n+13,5). - Zerinvary Lajos, Nov 25 2006
a(n) = A214292(n+13,6). - Reinhard Zumkeller, Jul 12 2012
From Amiram Eldar, Sep 06 2022: (Start)
Sum_{n>=0} 1/a(n) = 323171/88339680.
Sum_{n>=0} (-1)^n/a(n) = 7929257917/88339680 - 55552*log(2)/429. (End)

A094385 Triangle read by rows: T(n, k) = binomial(2n, k-1)binomial(2*n-k-1, n-k)/n for n, k >= 1, and T(n, 0) = 0^n.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 2, 6, 5, 0, 5, 20, 28, 14, 0, 14, 70, 135, 120, 42, 0, 42, 252, 616, 770, 495, 132, 0, 132, 924, 2730, 4368, 4004, 2002, 429, 0, 429, 3432, 11880, 23100, 27300, 19656, 8008, 1430, 0, 1430, 12870, 51051, 116688, 168300, 157080, 92820, 31824, 4862

Offset: 0

Philippe Deléham, Jun 03 2004, Jun 14 2007

			Triangle begins:
  1;
  0,   1;
  0,   1,    2;
  0,   2,    6,     5;
  0,   5,   20,    28,    14;
  0,  14,   70,   135,   120,    42;
  0,  42,  252,   616,   770,   495,   132;
  0, 132,  924,  2730,  4368,  4004,  2002,  429;
  0, 429, 3432, 11880, 23100, 27300, 19656, 8008, 1430; ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Paul Barry, A Note on a Family of Generalized Pascal Matrices Defined by Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.5.4.
Yue Cai and Catherine Yan, Counting with Borel's triangle, Texas A&M University.
Yue Cai and Catherine Yan, Counting with Borel's triangle, arXiv:1804.01597 [math.CO], 2018.
B. Derrida, E. Domany and D. Mukamel, A exact solution of a one-dimensional asymmetric exclusion model with open boundaries, J. Stat. Phys. 69, 1992, 667-687; eqs.(20), (21), p. 672.
Jian Zhou, On Some Mathematics Related to the Interpolating Statistics, arXiv:2108.10514 [math-ph], 2021.

Variant of A062991, unsigned and transposed.
See also A234950 for another version.
Columns: A000007 (k=0), 2*A001700 (k=1).
Diagonals: A002694 (k=n-1), A000108 (k=n).
Row sums: A064062 (generalized Catalan C(2; n)).
Cf. A000012, A002694, A064062, A064063, A064087, A064088, A064089.
Cf. A064090, A064091, A064092, A064093, A064094.

A094385:= func< n,k | n eq 0 select 1 else Binomial(2*n, k-1)*Binomial(2*n-k-1, n-k)/n >;
[A094385(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 26 2024

T[n_, k_] := Binomial[2n, k-1] Binomial[2n-k-1, n-k]/n; T[0, 0] = 1;
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 19 2018 *)

def A094385(n,k): return 1 if (n==0) else binomial(2*n,k-1)*binomial(2*n-k-1, n-k)//n
flatten([[A094385(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Sep 26 2024

T is given by [0, 1, 1, 1, 1, 1, 1, ...] DELTA [1, 1, 1, 1, 1, 1, 1, 1, 1, ...] where DELTA is the operator defined in A084938.
Sum_{k = 0..n} T(n, k)*x^(n-k) = C(x+1; n), generalized Catalan numbers; see left diagonals of triangle A064094: A000012, A000108, A064062, A064063, A064087..A064093 for x = -1, 0, ..., 9, respectively.
From G. C. Greubel, Sep 26 2024: (Start)
T(n, 1) = A000108(n-1), n >= 1.
T(n, n-1) = A002694(n), n >= 1.
T(n, n) = A000108(n). (End)

New name using a formula of the author by Peter Luschny, Sep 26 2024

A234950 Borel's triangle read by rows: T(n,k) = Sum_{s=k..n} binomial(s,k)*C(n,s), where C(n,s) is an entry in Catalan's triangle A009766.

Original entry on oeis.org

1, 2, 1, 5, 6, 2, 14, 28, 20, 5, 42, 120, 135, 70, 14, 132, 495, 770, 616, 252, 42, 429, 2002, 4004, 4368, 2730, 924, 132, 1430, 8008, 19656, 27300, 23100, 11880, 3432, 429, 4862, 31824, 92820, 157080, 168300, 116688, 51051, 12870, 1430

Offset: 0

N. J. A. Sloane, Jan 11 2014

			Triangle begins:
     1,
     2,    1,
     5,    6,     2,
    14,   28,    20,     5,
    42,  120,   135,    70,    14,
   132,  495,   770,   616,   252,    42,
   429, 2002,  4004,  4368,  2730,   924,  132,
  1430, 8008, 19656, 27300, 23100, 11880, 3432, 429,
  ...

Reinhard Zumkeller, Rows n=0..125 of triangle, flattened
Antoine Abram, Florent Hivert, James D. Mitchell, Jean-Christophe Novelli, and Maria Tsalakou, Power Quotients of Plactic-like Monoids, arXiv:2406.16387 [math.CO], 2024. See p. 5.
Paul Barry, Characterizations of the Borel triangle and Borel polynomials, arXiv:2001.08799 [math.CO], 2020.
Steve Butler, R. Graham, and C. H. Yan, Parking distributions on trees, European Journal of Combinatorics 65 (2017), 168-185.
Yue Cai and Catherine Yan, Counting with Borel's triangle, Texas A&M University.
Yue Cai and Catherine Yan, Counting with Borel's triangle, arXiv:1804.01597 [math.CO], 2018.
G. Chatel and V. Pilaud, Cambrian Hopf Algebras, arXiv:1411.3704 [math.CO], 2014-2015.
C. A. Francisco, J. Mermin, and J. Schweig, Catalan numbers, binary trees, and pointed pseudotriangulations, preprint 2013; European Journal of Combinatorics, Volume 45, April 2015, pp. 85-96.
Lord C. Kavi and Michael W. Newman, Counting closed walks in infinite regular trees using Catalan and Borel's triangles, arXiv:2212.08795 [math.CO], 2022.
A. Lakshminarayan, Z. Puchala, and K. Zyczkowski, Diagonal unitary entangling gates and contradiagonal quantum states, arXiv preprint arXiv:1407.1169 [quant-ph], 2014.
Jeffrey B. Remmel, Consecutive Up-down Patterns in Up-down Permutations, Electron. J. Combin., 21 (2014), #P3.2. See pp. 21-22. - _N. J. A. Sloane_, Jul 12 2014

A062991 is a signed version. See also A094385 for another version.
Cf. A009766.
The two borders give the Catalan numbers A000108.
Cf. A062992 (row sums).
The second and third columns give A002694 and A244887.

a234950 n k = sum [a007318 s k * a009766 n s | s <- [k..n]]
a234950_row n = map (a234950 n) [0..n]
a234950_tabl = map a234950_row [0..]
-- Reinhard Zumkeller, Jan 12 2014

T := (n,k) -> 2*binomial(2*n+1,n)*(n-k+1)*binomial(n+1,k)/((k+n+1)*(k+n+2)):
seq(seq(T(n,k), k=0..n), n=0..8); # Peter Luschny, Sep 04 2018

T[n_, k_] := 2 Binomial[2n+1, n] (n-k+1) Binomial[n+1, k]/((k+n+1)(k+n+2));
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 19 2018, from Maple *)

T(n,k) = sum(s=k, n, binomial(s, k)*binomial(n+s, n)*(n-s+1)/(n+1));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print();); \\ Michel Marcus, Sep 06 2015

G.f.: 1/x*(1-sqrt(1-4*x-4*x*y))/(1+2*y+sqrt(1-4*x-4*x*y)). - Vladimir Kruchinin, Sep 04 2018

T(n,k) = 2*binomial(2*n+1,n)*(n-k+1)*binomial(n+1,k)/((k+n+1)*(k+n+2)). - Peter Luschny, Sep 04 2018

A062746 Coefficient array for certain polynomials N(3; k,x) (rising powers of x).

Original entry on oeis.org

1, 3, -3, 1, 12, -29, 30, -15, 3, 55, -222, 405, -417, 252, -84, 12, 273, -1575, 4203, -6678, 6846, -4608, 1980, -495, 55, 1428, -10812, 38367, -83244, 121518, -124146, 89595, -44990, 15015, -3003, 273, 7752, -73017, 325164

Offset: 0

Wolfdieter Lang, Jul 12 2001

The g.f. for the sequence of column r=2*k+1, k >= 0, of array A062745(n,r) is N(3; k,x)*(x^(k+1))/(1-x)^(2*k+2) with N(3; k,x) := sum(a(k,p)*x^p,p=0..2*k).
The m=0 column gives: A001764(n+1). The row sums give A000012 (powers of 1) and (unsigned) A062747.
The sequence of step width of this staircase array is [1,2,2,2,...], i.e. the degree of the row polynomials is [0,2,4,6,...]= A005843.

			{1}; {3,-3,1}; {12,-29,30,-15,3}; ...; N(3; 1,x)= 3-3*x+x^2.

Cf. A062991.

a(k, p) := [x^p]N(3; k, x) with N(3; k, x)=(N(3; k-1, x)-A001764(k)*(1-x)^(2*k+1))/x, N(3; 0, x) := 1.

a(n, k)= a(n-1, k+1)+((-1)^k)*binomial(2*n+1, k+1)*binomial(3*n+1, n)/(3*n+1) if k=0, .., (2*n-3); a(n, k)= ((-1)^k)*binomial(2*n+1, k+1)*binomial(3*n+1, n)/(3*n+1) if k=(2*n-2), ..., 2*n; else 0.

cp's OEIS Frontend

A062992 Row sums of unsigned triangle A062991.

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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).

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A088617 Triangle read by rows: T(n,k) = C(n+k,n)*C(n,k)/(k+1), for n >= 0, k = 0..n.

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A005587 a(n) = n*(n+5)*(n+6)*(n+7)/24.

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A064059 Seventh column of Catalan triangle A009766.

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A005587 a(n) = n(n+5)(n+6)*(n+7)/24.

A094385 Triangle read by rows: T(n, k) = binomial(2n, k-1)binomial(2*n-k-1, n-k)/n for n, k >= 1, and T(n, 0) = 0^n.