A060693 -id:A060693 - cp's OEIS frontend

A006318 Large Schröder numbers (or large Schroeder numbers, or big Schroeder numbers).

1, 2, 6, 22, 90, 394, 1806, 8558, 41586, 206098, 1037718, 5293446, 27297738, 142078746, 745387038, 3937603038, 20927156706, 111818026018, 600318853926, 3236724317174, 17518619320890, 95149655201962, 518431875418926, 2832923350929742, 15521467648875090

Offset: 0

N. J. A. Sloane

For the little Schröder numbers (or little Schroeder numbers, or small Schroeder numbers) see A001003.
The number of perfect matchings in a triangular grid of n squares (n = 1, 4, 9, 16, 25, ...). - Roberto E. Martinez II, Nov 05 2001
a(n) is the number of subdiagonal paths from (0, 0) to (n, n) consisting of steps East (1, 0), North (0, 1) and Northeast (1, 1) (sometimes called royal paths). - David Callan, Mar 14 2004
Twice A001003 (except for the first term).
a(n) is the number of dissections of a regular (n+4)-gon by diagonals that do not touch the base. (A diagonal is a straight line joining two nonconsecutive vertices and dissection means the diagonals are noncrossing though they may share an endpoint. One side of the (n+4)-gon is designated the base.) Example: a(1)=2 because a pentagon has only 2 such dissections: the empty one and the one with a diagonal parallel to the base. - David Callan, Aug 02 2004
a(n) is the number of separable permutations, i.e., permutations avoiding 2413 and 3142 (see Shapiro and Stephens). - Vincent Vatter, Aug 16 2006
Eric W. Weisstein comments that the Schröder numbers bear the same relationship to the Delannoy numbers (A001850) as the Catalan numbers (A000108) do to the binomial coefficients. - Jonathan Vos Post, Dec 23 2004
a(n) is the number of lattice paths from (0, 0) to (n+1, n+1) consisting of unit steps north N = (0, 1) and variable-length steps east E = (k, 0), with k a positive integer, that stay strictly below the line y = x except at the endpoints. For example, a(2) = 6 counts 111NNN, 21NNN, 3NNN, 12NNN, 11N1NN, 2N1NN (east steps indicated by their length). If the word "strictly" is replaced by "weakly", the counting sequence becomes the little Schröder numbers, A001003 (offset). - David Callan, Jun 07 2006
a(n) is the number of dissections of a regular (n+3)-gon with base AB that do not contain a triangle of the form ABP with BP a diagonal. Example: a(1) = 2 because the square D-C | | A-B has only 2 such dissections: the empty one and the one with the single diagonal AC (although this dissection contains the triangle ABC, BC is not a diagonal). - David Callan, Jul 14 2006
a(n) is the number of (colored) Motzkin n-paths with each upstep and each flatstep at ground level getting one of 2 colors and each flatstep not at ground level getting one of 3 colors. Example: With their colors immediately following upsteps/flatsteps, a(2) = 6 counts U1D, U2D, F1F1, F1F2, F2F1, F2F2. - David Callan, Aug 16 2006
The Hankel transform of this sequence is A006125(n+1) = [1, 2, 8, 64, 1024, 32768, ...]; example: Det([1, 2, 6, 22; 2, 6, 22, 90; 6, 22, 90, 394; 22, 90, 394, 1806]) = 64. - Philippe Deléham, Sep 03 2006
Triangle A144156 has row sums equal to A006318 with left border A001003. - Gary W. Adamson, Sep 12 2008
a(n) is also the number of order-preserving and order-decreasing partial transformations (of an n-chain). Equivalently, it is the order of the Schröder monoid, PC sub n. - Abdullahi Umar, Oct 02 2008
Sum_{n >= 0} a(n)/10^n - 1 = (9 - sqrt(41))/2. - Mark Dols, Jun 22 2010
1/sqrt(41) = Sum_{n >= 0} Delannoy number(n)/10^n. - Mark Dols, Jun 22 2010
a(n) is also the dimension of the space Hoch(n) related to Hochschild two-cocycles. - Ph. Leroux (ph_ler_math(AT)yahoo.com), Aug 24 2010
Let W = (w(n, k)) denote the augmentation triangle (as at A193091) of A154325; then w(n, n) = A006318(n). - Clark Kimberling, Jul 30 2011
Conjecture: For each n > 2, the polynomial sum_{k = 0}^n a(k)*x^{n-k} is irreducible modulo some prime p < n*(n+1). - Zhi-Wei Sun, Apr 07 2013
From Jon Perry, May 24 2013: (Start)
Consider a Pascal triangle variant where T(n, k) = T(n, k-1) + T(n-1, k-1) + T(n-1, k), i.e., the order of performing the calculation must go from left to right (A033877). This sequence is the rightmost diagonal.
Triangle begins:
  1;
  1,  2;
  1,  4,  6;
  1,  6, 16, 22;
  1,  8, 30, 68, 90;
  ... (End)
a(n) is the number of permutations avoiding 2143, 3142 and one of the patterns among 246135, 254613, 263514, 524361, 546132. - Alexander Burstein, Oct 05 2014
a(n) is the number of semi-standard Young tableaux of shape n x 2 with consecutive entries.  That is, j in P and 1 <= i<= j imply i in P. - Graham H. Hawkes, Feb 15 2015
a(n) is the number of unary-rooted size n unary-binary trees (each node has either 1 or 2 degree out). - John Bodeen, May 29 2017
Conjecturally, a(n) is the number of permutations pi of length n such that s(pi) avoids the patterns 231 and 321, where s denotes West's stack-sorting map. - Colin Defant, Sep 17 2018
a(n) is the number of n X n permutation matrices which percolate under the 2-neighbor bootstrap percolation rule (see Shapiro and Stephens). The number of general n X n matrices of weight n which percolate is given in A146971. - Jonathan Noel, Oct 05 2018
a(n) is the number of permutations of length n+1 which avoid 3142 and 3241. The permutations are precisely the permutations that are sortable by a decreasing stack followed by an increasing stack in series. - Rebecca Smith, Jun 06 2019
a(n) is the number of permutations of length n+1 avoiding the partially ordered pattern (POP) {3>1, 4>1, 1>2} of length 4. That is, the number of length n+1 permutations having no subsequences of length 4 in which the second element is the smallest, and the first element is smaller than the third and fourth elements. - Sergey Kitaev, Dec 10 2020
Named after the German mathematician Ernst Schröder (1841-1902). - Amiram Eldar, Apr 15 2021
a(n) is the number of sequences of nonnegative integers (u_1, u_2, ..., u_n) such that (i) u_i <= i for all i, and (ii) the nonzero u_i are weakly increasing. For example, a(2) = 6 counts 00, 01, 02, 10, 11, 12. See link "Some bijections for lattice paths" at A001003. - David Callan, Dec 18 2021
a(n) is the number of separable elements of the Weyl group of type B_n/C_n (see Gaetz and Gao). - Fern Gossow, Jul 31 2023
The number of domino tilings of an Aztec triangle of order n. Dually, the number perfect matchings of the edges in the cellular graph formed by a triangular grid of n squares (n = 1, 4, 9, 16, 25, ...) as in Ciucu (1996). - Michael Somos, Sep 16 2024
a(n) is the number of dissections of a convex (n+3)-sided polygon by non-intersecting diagonals such that none of the dividing diagonals passes through a chosen vertex. - Muhammed Sefa Saydam, Mar 01 2025
a(n) is the number of dissections of a convex (n+m+1)-sided polygon by non-intersecting diagonals such that the selected m consecutive sides of the polygon will be in the same subpolygon. - Muhammed Sefa Saydam, Jul 02 2025

			a(3) = 22 since the top row of Q^n = (6, 6, 6, 4, 0, 0, 0, ...); where 22 = (6 + 6 + 6 + 4).
G.f. = 1 + 2*x + 6*x^2 + 22*x^3 + 90*x^4 + 394*x^5 + 1806*x^6 + 8858*x^7 + 41586*x^8 + ...

D. Andrica and E. J. Ionascu, On the number of polynomials with coefficients in [n], An. St. Univ. Ovidius Constanta, 2013, to appear.
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
Paul Barry, Riordan-Bernstein Polynomials, Hankel Transforms and Somos Sequences, Journal of Integer Sequences, Vol. 15 2012, #12.8.2.
Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
O. Bodini, A. Genitrini, F. Peschanski, and N.Rolin, Associativity for binary parallel processes, CALDAM 2015.
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, pages 24, 618.
S. Brlek, E. Duchi, E. Pergola, and S. Rinaldi, On the equivalence problem for succession rules, Discr. Math., 298 (2005), 142-154.
Xiang-Ke Chang, XB Hu, H Lei, and YN Yeh, Combinatorial proofs of addition formulas, The Electronic Journal of Combinatorics, 23(1) (2016), #P1.8.
William Y. C. Chen and Carol J. Wang, Noncrossing Linked Partitions and Large (3, 2)-Motzkin Paths, Discrete Math., 312 (2012), 1918-1922.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 81, #21, (4), q_n.
D. E. Davenport, L. W. Shapiro, and L. C. Woodson, The Double Riordan Group, The Electronic Journal of Combinatorics, 18(2) (2012), #P33.
Deng, Eva Y. P.; Dukes, Mark; Mansour, Toufik; and Wu, Susan Y. J.; Symmetric Schröder paths and restricted involutions. Discrete Math. 309 (2009), no. 12, 4108-4115. See p. 4109.
E. Deutsch, A bijective proof of an equation linking the Schroeder numbers, large and small, Discrete Math., 241 (2001), 235-240.
C. Domb and A. J. Barrett, Enumeration of ladder graphs, Discrete Math. 9 (1974), 341-358.
Doslic, Tomislav and Veljan, Darko. Logarithmic behavior of some combinatorial sequences. Discrete Math. 308 (2008), no. 11, 2182--2212. MR2404544 (2009j:05019) - From N. J. A. Sloane, May 01 2012
M. Dziemianczuk, Generalizing Delannoy numbers via counting weighted lattice paths, INTEGERS, 13 (2013), #A54.
Egge, Eric S., Restricted signed permutations counted by the Schröder numbers. Discrete Math. 306 (2006), 552-563. [Many applications of these numbers.]
S. Getu et al., How to guess a generating function, SIAM J. Discrete Math., 5 (1992), 497-499.
S. Gire, Arbres, permutations a motifs exclus et cartes planaire: quelques problemes algorithmiques et combinatoires, Ph.D. Thesis, Universite Bordeaux I, 1993.
N. S. S. Gu, N. Y. Li, and T. Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.
Guruswami, Venkatesan, Enumerative aspects of certain subclasses of perfect graphs. Discrete Math. 205 (1999), 97-117.
Silvia Heubach and Toufik Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.
D. E. Knuth, The Art of Computer Programming, Vol. 1, Section 2.2.1, Problem 11.
D. Kremer, Permutations with forbidden subsequences and a generalized Schröder number, Discrete Math. 218 (2000) 121-130.
Kremer, Darla and Shiu, Wai Chee; Finite transition matrices for permutations avoiding pairs of length four patterns. Discrete Math. 268 (2003), 171-183. MR1983276 (2004b:05006). See Table 1.
Laradji, A. and Umar, A. Asymptotic results for semigroups of order-preserving partial transformations. Comm. Algebra 34 (2006), 1071-1075. - Abdullahi Umar, Oct 11 2008
L. Moser and W. Zayachkowski, Lattice paths with diagonal steps, Scripta Math., 26 (1961), 223-229.
L. Shapiro and A. B. Stephens, Bootstrap percolation, the Schröder numbers and the N-kings problem, SIAM J. Discrete Math., Vol. 4 (1991), pp. 275-280.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see page 178 and also Problems 6.39 and 6.40.
Lin Yang and S.-L. Yang, The parametric Pascal rhombus. Fib. Q., 57:4 (2019), 337-346.
Sheng-Liang Yang and Mei-yang Jiang, The m-Schröder paths and m-Schröder numbers, Disc. Math. (2021) Vol. 344, Issue 2, 112209. doi:10.1016/j.disc.2020.112209. See Table 1.

Fung Lam, Table of n, a(n) for n = 0..2000 (terms 0..100 by T. D. Noe)
J. Abate and W. Whitt, Integer Sequences from Queueing Theory, J. Int. Seq. 13 (2010), 10.5.5, Corollary 8.
M. Aigner, Enumeration via ballot numbers, Discrete Math., 308 (2008), 2544-2563.
Andrei Asinowski and Cyril Banderier, From geometry to generating functions: rectangulations and permutations, arXiv:2401.05558 [cs.DM], 2024. See page 2.
Andrei Asinowski, G. Barequet, M. Bousquet-Mélou, T. Mansour, and R. Pinter, Orders induced by segments in floorplans and (2-14-3,3-41-2)-avoiding permutations, arXiv:1011.1889 [math.CO], 2010-2012.
M. D. Atkinson and T. Stitt, Restricted permutations and the wreath product, Preprint, 2002.
M. D. Atkinson and T. Stitt, Restricted permutations and the wreath product, Discrete Math., 259 (2002), 19-36.
Yu Hin Au, Some Properties and Combinatorial Implications of Weighted Small Schröder Numbers, arXiv:1912.00555 [math.CO], 2019.
Jean-Christophe Aval and F. Bergeron, Rectangular Schroder Parking Functions Combinatorics, arXiv:1603.09487 [math.CO], 2016.
Axel Bacher, Improving the Florentine algorithms: recovering algorithms for Motzkin and Schröder paths, arXiv:1802.06030 [cs.DS], 2018.
C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps, FPSAC02, Melbourne, 2002.
Kayleigh Bangs, Skye Binegar, Young Kim, Kyle Ormsby, Angélica M. Osorno, David Tamas-Parris, and Livia Xu, Biased permutative equivariant categories, arXiv:1907.00933 [math.AT], 2019.
E. Barcucci, A. Del Lungo, E. Pergola, and R. Pinzani, Permutations avoiding an increasing number of length-increasing forbidden subsequences, Discrete Mathematics and Theoretical Computer Science, 4 (2000), 31-44.
E. Barcucci, A. Del Lungo, E. Pergola, and R. Pinzani, Some permutations with forbidden subsequences and their inversion number, Discrete Math. 234(1-3) (2001), 1-15.
E. Barcucci, E. Pergola, R. Pinzani, and S. Rinaldi, ECO method and hill-free generalized Motzkin paths, Séminaire Lotharingien de Combinatoire, B46b (2001), 14 pp.
J.-L. Baril, C. Khalil, and V. Vajnovszki, Catalan and Schröder permutations sortable by two restricted stacks, arXiv:2004.01812 [cs.DM], 2020.
Marilena Barnabei, Flavio Bonetti, and Niccolò Castronuovo, Motzkin and Catalan Tunnel Polynomials, J. Int. Seq., Vol. 21 (2018), Article 18.8.8.
Paul Barry, Continued fractions and transformations of integer sequences, JIS 12 (2009) 09.7.6
Paul Barry, Generalized Catalan Numbers, Hankel Transforms and Somos-4 Sequences , J. Int. Seq. 13 (2010) #10.7.2.
Paul Barry, On a Generalization of the Narayana Triangle, J. Int. Seq. 14 (2011), Article 11.4.5.
Paul Barry, Laurent Biorthogonal Polynomials and Riordan Arrays, arXiv preprint arXiv:1311.2292 [math.CA], 2013.
Paul Barry, On a transformation of Riordan moment sequences, arXiv:1802.03443 [math.CO], 2018.
Paul Barry, Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles, J. Int. Seq., 22 (2019), Article 19.5.8.
Paul Barry, Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials, arXiv:1910.00875 [math.CO], 2019.
Paul Barry, Riordan arrays, the A-matrix, and Somos 4 sequences, arXiv:1912.01126 [math.CO], 2019.
Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See pp. 25, 29.
Paul Barry and Aoife Hennessy, A Note on Narayana Triangles and Related Polynomials, Riordan Arrays, and MIMO Capacity Calculations, J. Int. Seq. 14 (2011), Article 11.3.8.
Paul Barry and Nikolaos Pantelidis,On pseudo-involutions, involutions and quasi-involutions in the group of almost Riordan arrays, J Algebr Comb 54, 399-423 (2021). (appeared in its aerated form,i.e. 1,0,2,0,6,0,...)
Christian Bean, Émile Nadeau, and Henning Ulfarsson, Enumeration of Permutation Classes and Weighted Labelled Independent Sets, arXiv:1912.07503 [math.CO], 2019.
Arkady Berenstein, Vladimir Retakh, Christophe Reutenauer, and Doron Zeilberger, The Reciprocal of Sum_{n >= 0} a^n b^n for non-commuting a and b, Catalan numbers and non-commutative quadratic equations, arXiv preprint arXiv:1206.4225 [math.CO], 2012. - From _N. J. A. Sloane_, Nov 28 2012
F. R. Bernhart and N. J. A. Sloane, Emails, April-May 1994.
J. Bloom and A. Burstein, Egge triples and unbalanced Wilf-equivalence, arXiv:1410.0230 [math.CO], 2014.
O. Bodini, A. Genitrini, and F. Peschanski, The Combinatorics of Non-determinism, In proc. IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS'13), Leibniz International Proceedings in Informatics, pp 425-436, 2013.
Miklós Bóna, Cheyne Homberger, Jay Pantone, and Vince Vatter, Pattern-avoiding involutions: exact and asymptotic enumeration, arxiv:1310.7003 [math.CO], 2013-2014.
M. Bremner and S. Madariaga, Lie and Jordan products in interchange algebras, arXiv:1408.3069 [math.RA], 2014-2015.
M. Bremner and S. Madariaga, Permutation of elements in double semigroups, arXiv:1405.2889 [math.RA], 2014-2015.
R. Brignall, S. Huczynska, and V. Vatter, Simple permutations and algebraic generating functions, arXiv:math/0608391 [math.CO], 2006.
Marie-Louise Bruner and Martin Lackner, On the Likelihood of Single-Peaked Preferences, arXiv:1505.05852 [cs.GT], 2015.
Alexander Burstein, Sergi Elizalde, and Toufik Mansour, Restricted Dumont permutations, Dyck paths and noncrossing partitions, arXiv:math/0610234 [math.CO], 2006. See Theorem 3.5.
Alexander Burstein and J. Pantone, Two examples of unbalanced Wilf-equivalence, arXiv:1402.3842 [math.CO], 2014.
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
David Callan, An application of a bijection of Mansour, Deng, and Du, arXiv:1210.6455 [math.CO], 2012.
David Callan, A note on a bijection for Schröder permutations, arXiv:1602.05571 [math.CO], 2016.
David Callan and Toufik Mansour, Five subsets of permutations enumerated as weak sorting permutations, arXiv:1602.05182 [math.CO], 2016.
Hui-Qin Cao and Hao Pan, A Stern-type congruence for the Schröder numbers, arXiv:1512.06310 [math.NT], 2015.
Jean Cardinal, Vera Sacristán, and Rodrigo I. Silveira, A Note on Flips in Diagonal Rectangulations, arXiv:1712.07919 [math.CO], 2017.
F. Chapoton, F. Hivert, and J.-C. Novelli, A set-operad of formal fractions and dendriform-like sub-operads, arXiv:1307.0092 [math.CO], 2013.
F. Chapoton and S. Giraudo, Enveloping operads and bicoloured noncrossing configurations, arXiv:1310.4521 [math.CO], 2013.
W. Y. C. Chen, L. H. Liu, and C. J. Wang, Linked Partitions and Permutation Tableaux, arXiv:1305.5357 [math.CO], 2013.
Z. Chen and H. Pan, Identities involving weighted Catalan-Schroder and Motzkin Paths, arXiv:1608.02448 (2016), eq. (1.13), a=2, b=1.
Shane Chern, On 0012-avoiding inversion sequences and a Conjecture of Lin and Ma, arXiv:2006.04318 [math.CO], 2020.
J. Cigler, Hankel determinants of some polynomial sequences, 2012.
Johann Cigler, Christian Krattenthaler, Hankel determinants of linear combinations of moments of orthogonal polynomials, arXiv:2003.01676 [math.CO], 2020.
M. Ciucu, Perfect matchings of cellular graphs, J. Algebraic Combin., 5 (1996) 87-103.
CombOS - Combinatorial Object Server, Generate slicing floorplans
Sylvie Corteel, Megan A. Martinez, Carla D. Savage, and Michael Weselcouch, Patterns in Inversion Sequences I, arXiv:1510.05434 [math.CO], 2015
S. Crowley, Mellin and Laplace Integral Transforms Related to the Harmonic Sawtooth Map and a Diversion Into The Theory Of Fractal Strings, Number Theory, viXra:1202.0079, 2012.
R. De Castro, A. L. Ramírez, and J. L. Ramírez, Applications in Enumerative Combinatorics of Infinite Weighted Automata and Graphs, arXiv:1310.2449 [math.PR], 2013.
Colin Defant, Stack-sorting preimages of permutation classes, arXiv:1809.03123 [math.CO], 2018.
Colin Defant, Troupes, Cumulants, and Stack-Sorting, arXiv:2004.11367 [math.CO], 2020.
Phan Thuan Do, Thi Thu Huong Tran, and Vincent Vajnovszki, Exhaustive generation for permutations avoiding a (colored) regular sets of patterns, arXiv:1809.00742 [cs.DM], 2018.
B. Drake, An inversion theorem for labeled trees and some limits of areas under lattice paths (Example 1.6.7), A dissertation presented to the Faculty of the Graduate School of Arts and Sciences of Brandeis University.
D. Drake, Bijections from Weighted Dyck Paths to Schröder Paths, J. Int. Seq. 13 (2010), #10.9.2.
Rosena R. X. Du, Xiaojie Fan, and Yue Zhao, Enumeration on row-increasing tableaux of shape 2 X n, arXiv:1803.01590 [math.CO], 2018.
M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, arXiv preprint arXiv:1410.5747 [math.CO], 2014.
James East and Nicholas Ham, Lattice paths and submonoids of Z^2, arXiv:1811.05735 [math.CO], 2018.
Ömer Eğecioğlu, Collier Gaiser, and Mei Yin, Enumerating pattern-avoiding permutations by leading terms, arXiv:2309.15964 [math.CO], 2023.
S.-P. Eu and T.-S. Fu, A simple proof of the Aztec diamond problem, arXiv:math/0412041 [math.CO], 2004.
Luca Ferrari and Emanuele Munarini, Enumeration of edges in some lattices of paths, arXiv:1203.6792 [math.CO], 2012 and J. Int. Seq. 17 (2014), #14.1.5
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see p. 474.
Shishuo Fu, Z. Lin, and J. Zeng, Two new unimodal descent polynomials, arXiv:1507.05184 [math.CO], 2015-2019.
Shishuo Fu and Yaling Wang, Bijective recurrences concerning two Schröder triangles, arXiv:1908.03912 [math.CO], 2019.
Christian Gaetz and Yibo Gao, Separable elements in Weyl groups, arXiv:1905.09331 [math.CO], 2019.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
Olivier Gérard, Illustration of initial terms
Étienne Ghys, Quand beaucoup de courbes se rencontrent — Images des Mathématiques, CNRS, 2009.
Étienne Ghys, Intersecting curves, Amer. Math. Monthly, 120 (2013), 232-242.
Étienne Ghys, A Singular Mathematical Promenade, arXiv:1612.06373, 2016.
Samuele Giraudo, Operads from posets and Koszul duality, arXiv:1504.04529 [math.CO], 2015.
Samuele Giraudo, Pluriassociative algebras II: The polydendriform operad and related operads, arXiv:1603.01394 [math.CO], 2016.
Samuele Giraudo, Tree series and pattern avoidance in syntax trees, arXiv:1903.00677 [math.CO], 2019.
D. Gouyou-Beauchamps and B. Vauquelin, Deux propriétés combinatoires des nombres de Schröder, Theor. Inform. Appl., 22 (1988), 361-388.
Taras Goy and Mark Shattuck, Determinant identities for the Catalan, Motzkin and Schröder numbers, The Art of Discrete and Applied Mathematics, Vol. 7 (2024), #P1.09.
Li Guo and Jun Pei, Averaging algebras, Schröder numbers and rooted trees, arXiv:1401.7386 [math.RA], 2014.
Nils Haug, T. Prellberg, and G. Siudem, Scaling in area-weighted generalized Motzkin paths, arXiv:1605.09643 [cond-mat.stat-mech], 2016.
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.
Cheyne Homberger, Patterns in Permutations and Involutions: A Structural and Enumerative Approach, 1410.2657 [math.CO], 2014.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 159.
S. Kamioka, Laurent biorthogonal polynomials, q-Narayana polynomials and domino tilings of the Aztec diamonds, arXiv:1309.0268 [math.CO], 2013.
Sergey Kitaev and Jeffrey Remmel, Simple marked mesh patterns, arXiv:1201.1323 [math.CO], 2012.
S. Kitaev and J. Remmel, Quadrant Marked Mesh Patterns, J. Int. Seq. 15 (2012), #12.4.7
Laszlo Kozma and T. Saranurak, Binary search trees and rectangulations, arXiv:1603.08151 [cs.DS], 2016.
G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973).
G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)
G. Kreweras, Aires des chemins surdiagonaux et application à un problème économique, Cahiers du Bureau universitaire de recherche opérationnelle Série Recherche 24 (1976): 1-8. [Annotated scanned copy]
Nate Kube and Frank Ruskey, Sequences That Satisfy a(n-a(n))=0, Journal of Integer Sequences, 8 (2005), Article 05.5.5.
Guillaume Lample and François Charton, Deep Learning for Symbolic Mathematics, arXiv:1912.01412 [cs.SC], 2019.
A. Laradji and A. Umar 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.
Tamás Lengyel, On some p-adic properties and supercongruences of Delannoy and Schröder Numbers, Integers (2021) Vol. 21, #A86.
Philippe Leroux, Hochschild two-cocycles and the good triple (As,Hoch,Mag infinity), arXiv:0806.4093 [math.RA], 2008.
Huyile Liang, Yanni Pei, and Yi Wang, Analytic combinatorics of coordination numbers of cubic lattices, arXiv:2302.11856 [math.CO], 2023. See p. 5.
Huyile Liang, Jeffrey Remmel, and Sainan Zheng, Stieltjes moment sequences of polynomials, arXiv:1710.05795 [math.CO], 2017, see pp. 18-19.
Peter Luschny, The Lost Catalan Numbers And The Schröder Tableaux.
Megan A. Martinez and Carla D. Savage, Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations, arXiv:1609.08106 [math.CO], 2016 [Section 2.24].
Peter McCalla and Asamoah Nkwanta, Catalan and Motzkin Integral Representations, arXiv:1901.07092 [math.NT], 2019.
Arturo Merino and Torsten Mütze, Combinatorial generation via permutation languages. III. Rectangulations, arXiv:2103.09333 [math.CO], 2021.
J.-C. Novelli and J.-Y. Thibon, Hopf algebras and dendriform structures arising from parking functions, Fundamenta Mathematicae 193 (2007), no. 3, 189-241 (arXiv:math/0511200 [math.CO]).
Igor Pak, Complexity problems in enumerative combinatorics, arXiv:1803.06636 [math.CO], 2018.
P. Peart and W.-J. Woan, Generating Functions via Hankel and Stieltjes Matrices, J. Integer Seqs., Vol. 3 (2000), #00.2.1.
Jun Pei and Li Guo, Averaging algebras, Schröder numbers, rooted trees and operads, Journal of Algebraic Combinatorics, 42(1) (2015), 73-109; arXiv:1401.7386 [math.RA], 2014.
E. Pergola and R. A. Sulanke, Schröder Triangles, Paths and Parallelogram Polyominoes, J. Integer Sequences, 1 (1998), #98.1.7.
Feng Qi and B.-N. Guo, Some explicit and recursive formulas of the large and little Schröder numbers, Arab Journal of Mathematical Sciences, June 2016.
Feng Qi, Xiao-ting Shi, and Bai-Ni Guo, Integral representations of the large and little Schroder numbers, preprint, 2016.
Markus Saers, Dekai Wu, and Chris Quirk, On the Expressivity of Linear Transductions, The 13th Machine Translation Summit.
Seunghyun Seo, The Catalan Threshold Arrangement, Journal of Integer Sequences, 20 (2017), #17.1.1.
L. W. Shapiro and N. J. A. Sloane, Correspondence, 1976.
P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26 (1978), 261-272.
P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers [Corrected annotated scanned copy]
R. A. Sulanke, Moments of generalized Motzkin paths, J. Integer Sequences, Vol. 3 (2000), #00.1.
R. A. Sulanke, Moments, Narayana numbers and the cut and paste for lattice paths.
R. A. Sulanke, Bijective recurrences concerning Schröder paths, Electron. J. Combin. 5 (1998), Research Paper 47, 11 pp.
Hua Sun and Yi Wang, A Combinatorial Proof of the Log-Convexity of Catalan-Like Numbers, J. Int. Seq. 17 (2014), #14.5.2
Zhi-Wei Sun, On Delannoy numbers and Schröder numbers, Journal of Number Theory, 131(12), (2011), 2387-2397; doi:10.1016/j.jnt.2011.06.005; arXiv 1009.2486 [math.NT].
Zhi-Wei Sun, Conjectures involving combinatorial sequences, arXiv:1208.2683 [math.CO], 2012. - _N. J. A. Sloane_, Dec 25 2012
Z.-W. Sun, Conjectures involving arithmetical sequences, Number Theory: Arithmetic in Shangri-La (eds., S. Kanemitsu, H.-Z. Li and J.-Y. Liu), Proc. the 6th China-Japan Sem. Number Theory (Shanghai, August 15-17, 2011), World Sci., Singapore, 2013, pp. 244-258. - _N. J. A. Sloane_, Dec 28 2012
Paul Tarau, On Type-directed Generation of Lambda Terms, preprint, 2015.
Paul Tarau, On logic programming representations of lambda terms: de Bruijn indices, compression, type inference, combinatorial generation, normalization, 2015.
Paul Tarau, A Logic Programming Playground for Lambda Terms, Combinators, Types and Tree-based Arithmetic Computations, arXiv:1507.06944 [cs.LO], 2015.
Paul Tarau, A Hiking Trip Through the Orders of Magnitude: Deriving Efficient Generators for Closed Simply-Typed Lambda Terms and Normal Forms, arXiv preprint arXiv:1608.03912 [cs.PL], 2016.
V. K. Varma and H. Monien, Renormalization of two-body interactions due to higher-body interactions of lattice bosons, arXiv:1211.5664 [cond-mat.quant-gas], 2012. - _N. J. A. Sloane_, Jan 03 2013
Vincent Vatter, Permutation classes, arXiv:1409.5159 [math.CO], 2014.
Yi Wang and Bao-Xuan Zhu, Proofs of some conjectures on monotonicity of number-theoretic and combinatorial sequences, arXiv:1303.5595 [math.CO], 2013.
M. S. Waterman, Home Page (contains copies of his papers)
Eric Weisstein's World of Mathematics, Schröder Number.
J. West, Generating trees and the Catalan and Schröder numbers, Discrete Math. 146 (1995), 247-262.
J. Winter, M. M. Bonsangue, and J. J. M. M. Rutten, Context-free coalgebras, 2013.
Wikipedia, Schröder number
Chunyan Yan and Zhicong Lin, Inversion sequences avoiding pairs of patterns, arXiv:1912.03674 [math.CO], 2019.
S.-n. Zheng and S.-l. Yang, On the-Shifted Central Coefficients of Riordan Matrices, Journal of Applied Mathematics 2014, Article ID 848374.
Index entries for "core" sequences

Apart from leading term, twice A001003 (the small Schroeder numbers). Cf. A025240.
Sequences A085403, A086456, A103137, A112478 are essentially the same sequence.
Main diagonal of A033877.
Cf. A002003, A004148, A088617, A060693, A144156.
Row sums of A104219. Bisections give A138462, A138463.
Row sums of A175124.
The sequences listed in Yang-Jiang's Table 1 appear to be A006318, A001003, A027307, A034015, A144097, A243675, A260332, A243676. - N. J. A. Sloane, Mar 28 2021

Concatenation([1],List([1..25],n->(1/n)*Sum([0..n],k->2^k*Binomial(n,k)*Binomial(n,k-1)))); # Muniru A Asiru, Nov 29 2018

a006318 n = a004148_list !! n
a006318_list = 1 : f [1] where
   f xs = y : f (y : xs) where
     y = head xs + sum (zipWith (*) xs $ reverse xs)
-- Reinhard Zumkeller, Nov 13 2012

Order := 24: solve(series((y-y^2)/(1+y),y)=x,y); # then A(x)=y(x)/x
BB:=(-1-z-sqrt(1-6*z+z^2))/2: BBser:=series(BB, z=0, 24): seq(coeff(BBser, z, n), n=1..23); # Zerinvary Lajos, Apr 10 2007
A006318_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := 2*a[w-1]+add(a[j]*a[w-j-1], j=1..w-1) od; convert(a,list)end: A006318_list(22); # Peter Luschny, May 19 2011
A006318 := n-> add(binomial(n+k, n-k) * binomial(2*k, k)/(k+1), k=0..n): seq(A006318(n), n=0..22); # Johannes W. Meijer, Jul 14 2013
seq(simplify(hypergeom([-n,n+1],[2],-1)), n=0..100); # Robert Israel, Mar 23 2015

a[0] = 1; a[n_Integer] := a[n] = a[n - 1] + Sum[a[k]*a[n - 1 - k], {k, 0, n - 1}]; Array[a[#] &, 30]
InverseSeries[Series[(y - y^2)/(1 + y), {y, 0, 24}], x] (* then A(x) = y(x)/x *) (* Len Smiley, Apr 11 2000 *)
CoefficientList[Series[(1 - x - (1 - 6x + x^2)^(1/2))/(2x), {x, 0, 30}], x] (* Harvey P. Dale, May 01 2011 *)
a[ n_] := 2 Hypergeometric2F1[ -n + 1, n + 2, 2, -1]; (* Michael Somos, Apr 03 2013 *)
a[ n_] := With[{m = If[ n < 0, -1 - n, n]}, SeriesCoefficient[(1 - x - Sqrt[ 1 - 6 x + x^2])/(2 x), {x, 0, m}]]; (* Michael Somos, Jun 10 2015 *)
Table[-(GegenbauerC[n+1, -1/2, 3] + KroneckerDelta[n])/2, {n, 0, 30}] (* Vladimir Reshetnikov, Nov 12 2016 *)
CoefficientList[Nest[1+x(#+#^2)&, 1+O[x], 20], x] (* Oliver Seipel, Dec 21 2024 *)

{a(n) = if( n<0, n = -1-n); polcoeff( (1 - x - sqrt( 1 - 6*x + x^2 + x^2 * O(x^n))) / 2, n+1)}; /* Michael Somos, Apr 03 2013 */

{a(n) = if( n<1, 1, sum( k=0, n, 2^k * binomial( n, k) * binomial( n, k-1)) / n)};

from gmpy2 import divexact
A006318 = [1, 2]
for n in range(3,10**3):
    A006318.append(int(divexact(A006318[-1]*(6*n-9)-(n-3)*A006318[-2],n)))
# Chai Wah Wu, Sep 01 2014

# Generalized algorithm of L. Seidel
def A006318_list(n) :
    D = [0]*(n+1); D[1] = 1
    b = True; h = 1; R = []
    for i in range(2*n) :
        if b :
            for k in range(h,0,-1) : D[k] += D[k-1]
            h += 1;
        else :
            for k in range(1,h, 1) : D[k] += D[k-1]
            R.append(D[h-1]);
        b = not b
    return R
A006318_list(23) # Peter Luschny, Jun 02 2012

G.f.: (1 - x - (1 - 6*x + x^2)^(1/2))/(2*x).
a(n) = 2*hypergeom([-n+1, n+2], [2], -1). - Vladeta Jovovic, Apr 24 2003
For n > 0, a(n) = (1/n)*Sum_{k = 0..n} 2^k*C(n, k)*C(n, k-1). - Benoit Cloitre, May 10 2003
The g.f. satisfies (1 - x)*A(x) - x*A(x)^2 = 1. - Ralf Stephan, Jun 30 2003
For the asymptotic behavior, see A001003 (remembering that A006318 = 2*A001003). - N. J. A. Sloane, Apr 10 2011
From Philippe Deléham, Nov 28 2003: (Start)
Row sums of A088617 and A060693.
a(n) = Sum_{k = 0..n} C(n+k, n)*C(n, k)/(k+1). (End)
With offset 1: a(1) = 1, a(n) = a(n-1) + Sum_{i = 1..n-1} a(i)*a(n-i). - Benoit Cloitre, Mar 16 2004
a(n) = Sum_{k = 0..n} A000108(k)*binomial(n+k, n-k). - Benoit Cloitre, May 09 2004
a(n) = Sum_{k = 0..n} A011117(n, k). - Philippe Deléham, Jul 10 2004
a(n) = (CentralDelannoy(n+1) - 3 * CentralDelannoy(n))/(2*n) = (-CentralDelannoy(n+1) + 6 * CentralDelannoy(n) - CentralDelannoy(n-1))/2 for n >= 1, where CentralDelannoy is A001850. - David Callan, Aug 16 2006
From Abdullahi Umar, Oct 11 2008: (Start)
A123164(n+1) - A123164(n) = (2*n+1)*a(n) (n >= 0).
  and 2*A123164(n) = (n+1)*a(n) - (n-1)*a(n-1) (n > 0). (End)
Define the general Delannoy numbers d(i, j) as in A001850. Then a(k) = d(2*k, k) - d(2*k, k-1) and a(0) = 1, Sum_{j=0..n} ((-1)^j * (d(n, j) + d(n-1, j-1)) * a(n-j)) = 0. - Peter E John, Oct 19 2006
Given an integer t >= 1 and initial values u = [a_0, a_1, ..., a_{t-1}], we may define an infinite sequence Phi(u) by setting a_n = a_{n-1} + a_0*a_{n-1} + a_1*a_{n-2} + ... + a_{n-2}*a_1 for n >= t. For example, Phi([1]) is the Catalan numbers A000108. The present sequence is (essentially) Phi([2]). - Gary W. Adamson, Oct 27 2008
G.f.: 1/(1-2x/(1-x/(1-2x/(1-x/(1-2x/(1-x/(1-2x/(1-x/(1-2x/(1-x.... (continued fraction). - Paul Barry, Dec 08 2008
G.f.: 1/(1 - x - x/(1 - x - x/(1 - x - x/(1 - x - x/(1 - x - x/(1 - ... (continued fraction). - Paul Barry, Jan 29 2009
a(n) ~ ((3 + 2*sqrt(2))^n)/(n*sqrt(2*Pi*n)*sqrt(3*sqrt(2) - 4))*(1-(9*sqrt(2) + 24)/(32*n) + ...). - G. Nemes (nemesgery(AT)gmail.com), Jan 25 2009
Logarithmic derivative yields A002003. - Paul D. Hanna, Oct 25 2010
a(n) = the upper left term in M^(n+1), M = the production matrix:
  1, 1, 0, 0, 0, 0, ...
  1, 1, 1, 0, 0, 0, ...
  2, 2, 1, 1, 0, 0, ...
  4, 4, 2, 1, 1, 0, ...
  8, 8, 8, 2, 1, 1, ...
  ... - Gary W. Adamson, Jul 08 2011
a(n) is the sum of top row terms in Q^n, Q = an infinite square production matrix as follows:
  1, 1, 0, 0, 0, 0, ...
  1, 1, 2, 0, 0, 0, ...
  1, 1, 1, 2, 0, 0, ...
  1, 1, 1, 1, 2, 0, ...
  1, 1, 1, 1, 1, 2, ...
  ... - Gary W. Adamson, Aug 23 2011
From Tom Copeland, Sep 21 2011: (Start)
With F(x) = (1 - 3*x - sqrt(1 - 6*x + x^2))/(2*x) an o.g.f. (nulling the n = 0 term) for A006318, G(x) = x/(2 + 3*x + x^2) is the compositional inverse.
Consequently, with H(x) = 1/ (dG(x)/dx) = (2 + 3*x + x^2)^2 / (2 - x^2),
  a(n) = (1/n!)*[(H(x)*d/dx)^n] x evaluated at x = 0, i.e.,
  F(x) = exp[x*H(u)*d/du] u, evaluated at u = 0. Also, dF(x)/dx = H(F(x)). (End)
a(n-1) = number of ordered complete binary trees with n leaves having k internal vertices colored black, the remaining n - 1 - k internal vertices colored white, and such that each vertex and its rightmost child have different colors ([Drake, Example 1.6.7]). For a refinement of this sequence see A175124. - Peter Bala, Sep 29 2011
D-finite with recurrence: (n-2)*a(n-2) - 3*(2*n-1)*a(n-1) + (n+1)*a(n) = 0. - Vaclav Kotesovec, Oct 05 2012
G.f.: A(x) = (1 - x - sqrt(1 - 6*x + x^2))/(2*x) = (1 - G(0))/x; G(k) = 1 + x - 2*x/G(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Jan 04 2012
G.f.: A(x) = (1 - x - sqrt(1 - 6*x + x^2))/(2*x) = (G(0) - 1)/x; G(k) = 1 - x/(1 - 2/G(k+1)); (continued fraction, 2-step). - Sergei N. Gladkovskii, Jan 04 2012
a(n+1) = a(n) + Sum_{k=0..n} a(k)*(n-k). - Reinhard Zumkeller, Nov 13 2012
G.f.: 1/Q(0) where Q(k) = 1 + k*(1 - x) - x - x*(k+1)*(k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Mar 14 2013
a(-1-n) = a(n). - Michael Somos, Apr 03 2013
G.f.: 1/x - 1 - U(0)/x, where U(k) = 1 - x - x/U(k+1); (continued fraction). - Sergei N. Gladkovskii, Jul 16 2013
G.f.: (2 - 2*x - G(0))/(4*x), where G(k) = 1 + 1/( 1 - x*(6 - x)*(2*k - 1)/(x*(6 - x)*(2*k - 1) + 2*(k + 1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 16 2013
a(n) = 1/(n + 1) * (Sum_{j=0..n} C(n+j, j)*C(n+j+1, j+1)*(Sum_{k=0..n-j} (-1)^k*C(n+j+k, k))). - Graham H. Hawkes, Feb 15 2015
a(n) = hypergeom([-n, n+1], [2], -1). - Peter Luschny, Mar 23 2015
a(n) = sqrt(2) * LegendreP(n, -1, 3) where LegendreP is the associated Legendre function of the first kind (in Maple's notation). - Robert Israel, Mar 23 2015
G.f. A(x) satisfies: A(x) = Sum_{j>=0} x^j * Sum_{k=0..j} binomial(j,k)*A(x)^k. - Ilya Gutkovskiy, Apr 11 2019
From Peter Bala, May 13 2024: (Start)
a(n) = 2 * Sum_{k = 0..floor(n/2)} binomial(n, 2*k)*binomial(2*n-2*k, n)/(n-2*k+1) for n >= 1.
a(n) = Integral_{x = 0..1} Legendre_P(n, 2*x+1) dx. (End)
G.f. A(x) = 1/(1 - x) * c(x/(1-x)^2), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. - Peter Bala, Aug 29 2024

Edited by Charles R Greathouse IV, Apr 20 2010

A090181 Triangle of Narayana (A001263) with 0 <= k <= n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 6, 1, 0, 1, 10, 20, 10, 1, 0, 1, 15, 50, 50, 15, 1, 0, 1, 21, 105, 175, 105, 21, 1, 0, 1, 28, 196, 490, 490, 196, 28, 1, 0, 1, 36, 336, 1176, 1764, 1176, 336, 36, 1, 0, 1, 45, 540, 2520, 5292, 5292, 2520, 540, 45, 1, 0, 1, 55, 825, 4950, 13860

Offset: 0

Philippe Deléham, Jan 19 2004

Number of Dyck n-paths with exactly k peaks. - Peter Luschny, May 10 2014

			Triangle starts:
[0] 1;
[1] 0, 1;
[2] 0, 1,  1;
[3] 0, 1,  3,   1;
[4] 0, 1,  6,   6,    1;
[5] 0, 1, 10,  20,   10,    1;
[6] 0, 1, 15,  50,   50,   15,    1;
[7] 0, 1, 21, 105,  175,  105,   21,   1;
[8] 0, 1, 28, 196,  490,  490,  196,  28,  1;
[9] 0, 1, 36, 336, 1176, 1764, 1176, 336, 36, 1;

Indranil Ghosh, Rows 0..100, flattened
Yu Hin Au, Some Properties and Combinatorial Implications of Weighted Small Schröder Numbers, arXiv:1912.00555 [math.CO], 2019.
Paul Barry, Continued fractions and transformations of integer sequences, JIS 12 (2009) 09.7.6.
Paul Barry, Invariant number triangles, eigentriangles and Somos-4 sequences, arXiv preprint arXiv:1107.5490 [math.CO], 2011.
Paul Barry, On a Generalization of the Narayana Triangle, J. Int. Seq. 14 (2011) # 11.4.5.
Paul Barry, On a transformation of Riordan moment sequences, arXiv:1802.03443 [math.CO], 2018.
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018.
Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See pp. 25, 29.
Paul Barry and Aoife Hennessy, A Note on Narayana Triangles and Related Polynomials, Riordan Arrays, and MIMO Capacity Calculations, J. Int. Seq. 14 (2011), Article 11.3.8.
FindStat - Combinatorial Statistic Finder, The number of peaks of a Dyck path, The number of double rises of a Dyck path, The number of valleys of a Dyck path, The number of left oriented leafs except the first one of a binary tree, The number of left tunnels of a Dyck path
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.

Mirror image of triangle A131198. A000108 (row sums, Catalan).
Columns: A000217, A002415, A006542, A006857.
Cf. A001263, A084938, A243752.
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A000108(n), A006318(n), A047891(n+1), A082298(n), A082301(n), A082302(n), A082305(n), A082366(n), A082367(n) for x=0,1,2,3,4,5,6,7,8,9. - Philippe Deléham, Aug 10 2006
Sum_{k=0..n} x^(n-k)*T(n,k) = A090192(n+1), A000012(n), 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, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. - Philippe Deléham, Oct 21 2006
Sum_{k=0..n} T(n,k)*x^k*(x-1)^(n-k) = A000012(n), A006318(n), A103210(n), A103211(n), A133305(n), A133306(n), A133307(n), A133308(n), A133309(n) for x = 1, 2, 3, 4, 5, 6, 7, 8, 9, respectively. - Philippe Deléham, Oct 20 2007

[[(&+[(-1)^(j-k)*Binomial(2*n-j,j)*Binomial(j,k)*Binomial(2*n-2*j,n-j)/(n-j+1): j in [0..n]]): k in [0..n]]: n in [0..10]];

A090181 := (n,k) -> binomial(n,n-k)*binomial(n-1,n-k)/(n-k+1):
seq(print( seq(A090181(n,k),k=0..n)),n=0..5); # Peter Luschny, May 10 2014
egf := 1+int((sqrt(t)*exp((1+t)*x)*BesselI(1,2*sqrt(t)*x))/x,x);
s := n -> n!*coeff(series(egf,x,n+2),x,n);
seq(print(seq(coeff(s(n),t,j),j=0..n)),n=0..9); # Peter Luschny, Oct 30 2014
T := proc(n, k) option remember; if k = n or k = 1 then 1 elif k < 1 then 0 else (2*n/k - 1) * T(n-1, k-1) + T(n-1, k) fi end:
for n from 0 to 8 do seq(T(n, k), k = 0..n) od;  # Peter Luschny, Dec 31 2024

Flatten[Table[Sum[(-1)^(j-k) * Binomial[2n-j,j] * Binomial[j,k] * CatalanNumber[n-j], {j, 0, n}], {n,0,11},{k,0,n}]] (* Indranil Ghosh, Mar 05 2017 *)
p[0, ] := 1; p[1, x] := x; p[n_, x_] := ((2 n - 1) (1 + x) p[n - 1, x] - (n - 2) (x - 1)^2 p[n - 2, x]) / (n + 1);
Table[CoefficientList[p[n, x], x], {n, 0, 9}] // TableForm (* Peter Luschny, Apr 26 2022 *)

c(n) = binomial(2*n,n)/ (n+1);
tabl(nn) = {for(n=0, nn, for(k=0, n, print1(sum(j=0, n, (-1)^(j-k) * binomial(2*n-j,j) * binomial(j,k) * c(n-j)),", ");); print(););};
tabl(11); \\ Indranil Ghosh, Mar 05 2017

from functools import cache
@cache
def Trow(n):
    if n == 0: return [1]
    if n == 1: return [0, 1]
    if n == 2: return [0, 1, 1]
    A = Trow(n - 2) + [0, 0]
    B = Trow(n - 1) + [1]
    for k in range(n - 1, 1, -1):
        B[k] = (((B[k] + B[k - 1]) * (2 * n - 1)
               - (A[k] - 2 * A[k - 1] + A[k - 2]) * (n - 2)) // (n + 1))
    return B
for n in range(10): print(Trow(n)) # Peter Luschny, May 02 2022

def A090181_row(n):
    U = [0]*(n+1)
    for d in DyckWords(n):
        U[d.number_of_peaks()] +=1
    return U
for n in range(8): A090181_row(n) # Peter Luschny, May 10 2014

Triangle T(n, k), read by rows, given by [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] where DELTA is the operator defined in A084938. T(0, 0) = 1, T(n, 0) = 0 for n>0, T(n, k) = C(n-1, k-1)*C(n, k-1)/k for k>0.
Sum_{j>=0} T(n,j)*binomial(j,k) = A060693(n,k). - Philippe Deléham, May 04 2007
Sum_{k=0..n} T(n,k)*10^k = A143749(n+1). - Philippe Deléham, Oct 14 2008
From Paul Barry, Nov 10 2008: (Start)
Coefficient array of the polynomials P(n,x) = x^n*2F1(-n,-n+1;2;1/x).
T(n,k) = Sum_{j=0..n} (-1)^(j-k)*C(2n-j,j)*C(j,k)*A000108(n-j). (End)
Sum_{k=0..n} T(n,k)*5^k*3^(n-k) = A152601(n). - Philippe Deléham, Dec 10 2008
Sum_{k=0..n} T(n,k)*(-2)^k = A152681(n); Sum_{k=0..n} T(n,k)*(-1)^k = A105523(n). - Philippe Deléham, Feb 03 2009
Sum_{k=0..n} T(n,k)*2^(n+k) = A156017(n). - Philippe Deléham, Nov 27 2011
T(n, k) = C(n,n-k)*C(n-1,n-k)/(n-k+1). - Peter Luschny, May 10 2014
E.g.f.: 1+Integral((sqrt(t)*exp((1+t)*x)*BesselI(1,2*sqrt(t)*x))/x dx). - Peter Luschny, Oct 30 2014
G.f.: (1+x-x*y-sqrt((1-x*(1+y))^2-4*y*x^2))/(2*x). - Alois P. Heinz, Nov 28 2021, edited by Ron L.J. van den Burg, Dec 19 2021
T(n, k) = [x^k] (((2*n - 1)*(1 + x)*p(n-1, x) - (n - 2)*(x - 1)^2*p(n-2, x))/(n + 1)) with p(0, x) = 1 and p(1, x) = x. - Peter Luschny, Apr 26 2022
Recursion based on rows (see the Python program):
T(n, k) = (((B(k) + B(k-1))*(2*n - 1) - (A(k) - 2*A(k-1) + A(k-2))*(n-2))/(n+1)), where A(k) = T(n-2, k) and B(k) = T(n-1, k), for n >= 3. # Peter Luschny, May 02 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

A033282 Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.

Original entry on oeis.org

1, 1, 2, 1, 5, 5, 1, 9, 21, 14, 1, 14, 56, 84, 42, 1, 20, 120, 300, 330, 132, 1, 27, 225, 825, 1485, 1287, 429, 1, 35, 385, 1925, 5005, 7007, 5005, 1430, 1, 44, 616, 4004, 14014, 28028, 32032, 19448, 4862, 1, 54, 936, 7644, 34398, 91728, 148512, 143208, 75582, 16796

Offset: 3

N. J. A. Sloane

T(n+3, k) is also the number of compatible k-sets of cluster variables in Fomin and Zelevinsky's cluster algebra of finite type A_n. Take a row of this triangle regarded as a polynomial in x and rewrite as a polynomial in y := x+1. The coefficients of the polynomial in y give a row of the triangle of Narayana numbers A001263. For example, x^2 + 5*x + 5 = y^2 + 3*y + 1. - Paul Boddington, Mar 07 2003
Number of standard Young tableaux of shape (k+1,k+1,1^(n-k-3)), where 1^(n-k-3) denotes a sequence of n-k-3 1's (see the Stanley reference).
Number of k-dimensional 'faces' of the n-dimensional associahedron (see Simion, p. 168). - Mitch Harris, Jan 16 2007
Mirror image of triangle A126216. - Philippe Deléham, Oct 19 2007
For relation to Lagrange inversion or series reversion and the geometry of associahedra or Stasheff polytopes (and other combinatorial objects) see A133437. - Tom Copeland, Sep 29 2008
Row generating polynomials 1/(n+1)*Jacobi_P(n,1,1,2*x+1). Row n of this triangle is the f-vector of the simplicial complex dual to an associahedron of type A_n [Fomin & Reading, p. 60]. See A001263 for the corresponding array of h-vectors for associahedra of type A_n. See A063007 and A080721 for the f-vectors for associahedra of type B and type D respectively. - Peter Bala, Oct 28 2008
f-vectors of secondary polytopes for Grobner bases for optimization and integer programming (see De Loera et al. and Thomas). - Tom Copeland, Oct 11 2011
From Devadoss and O'Rourke's book: The Fulton-MacPherson compactification of the configuration space of n free particles on a line segment with a fixed particle at each end is the n-Dim Stasheff associahedron whose refined f-vector is given in A133437 which reduces to A033282. - Tom Copeland, Nov 29 2011
Diagonals of A132081 are rows of A033282. - Tom Copeland, May 08 2012
The general results on the convolution of the refined partition polynomials of A133437, with u_1 = 1 and u_n = -t otherwise, can be applied here to obtain results of convolutions of these polynomials. - Tom Copeland, Sep 20 2016
The signed triangle t(n, k) =(-1)^k* T(n+2, k-1), n >= 1, k = 1..n, seems to be obtainable from the partition array A111785 (in Abramowitz-Stegun order) by adding the entries corresponding to the partitions of n with the number of parts k. E.g., triangle t, row n=4: -1, (6+3) = 9, -21, 14. - Wolfdieter Lang, Mar 17 2017
The preceding conjecture by Lang is true. It is implicit in Copeland's 2011 comments in A086810 on the relations among a gf and its compositional inverse for that entry and inversion through A133437 (a differently normalized version of A111785), whose integer partitions are the same as those for A134685. (An inversion pair in Copeland's 2008 formulas below can also be used to prove the conjecture.) In addition, it follows from the relation between the inversion formula of A111785/A133437 and the enumeration of distinct faces of associahedra. See the MathOverflow link concernimg Loday and the Aguiar and Ardila reference in A133437 for proofs of the relations between the partition polynomials for inversion and enumeration of the distinct faces of the A_n associahedra, or Stasheff polytopes. - Tom Copeland, Dec 21 2017
The rows seem to give (up to sign) the coefficients in the expansion of the integer-valued polynomial (x+1)*(x+2)^2*(x+3)^2*...*(x+n)^2*(x+n+1)/(n!*(n+1)!) in the basis made of the binomial(x+i,i). - F. Chapoton, Oct 07 2022
Chapoton's observation above is correct: the precise expansion is (x+1)*(x+2)^2*(x+3)^2*...*(x+n)^2*(x+n+1)/ (n!*(n+1)!) = Sum_{k = 0..n-1} (-1)^k*T(n+2,n-k-1)*binomial(x+2*n-k,2*n-k), as can be verified using the WZ algorithm. For example, n = 4 gives (x+1)*(x+2)^2*(x+3)^2*(x+4)^2*(x+5)/(4!*5!) = 14*binomial(x+8,8) - 21*binomial(x+7,7) + 9*binomial(x+6,6) - binomial(x+5,5). - Peter Bala, Jun 24 2023

			The triangle T(n, k) begins:
n\k  0  1   2    3     4     5      6      7     8     9
3:   1
4:   1  2
5:   1  5   5
6:   1  9  21   14
7:   1 14  56   84    42
8:   1 20 120  300   330   132
9:   1 27 225  825  1485  1287    429
10:  1 35 385 1925  5005  7007   5005   1430
11:  1 44 616 4004 14014 28028  32032  19448  4862
12:  1 54 936 7644 34398 91728 148512 143208 75582 16796
... reformatted. - _Wolfdieter Lang_, Mar 17 2017

S. Devadoss and J. O'Rourke, Discrete and Computational Geometry, Princeton Univ. Press, 2011 (See p. 241.)
Ronald L. Graham, Donald E. Knuth, Oren Patashnik, Concrete Mathematics, 2nd ed., Addison-Wesley, 1994. Exercise 7.50, pages 379, 573.
T. K. Petersen, Eulerian Numbers, Birkhauser, 2015, Section 5.8.

Vincenzo Librandi, Table of n, a(n) for n = 3..2000
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, 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, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
Paul Barry, On the f-Matrices of Pascal-like Triangles Defined by Riordan Arrays, arXiv:1805.02274 [math.CO], 2018.
Paul Barry, Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.
Karin Baur and P. P. Martin, The fibres of the Scott map on polygon tilings are the flip equivalence classes, arXiv:1601.05080 [math.CO], 2016.
David Beckwith, Legendre polynomials and polygon dissections?, Amer. Math. Monthly, 105 (1998), 256-257.
William Butler, Andrew Kalotay and N. J. A. Sloane, Correspondence, 1974
Arthur Cayley, On the partitions of a polygon, Proc. London Math. Soc., 22 (1891), 237-262 = Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 93ff. (See p. 239.)
Adrian Celestino and Yannic Vargas, Schröder trees, antipode formulas and non-commutative probability, arXiv:2311.07824 [math.CO], 2023.
Frédéric Chapoton, Enumerative properties of generalized associahedra, Séminaire Lotharingien de Combinatoire, B51b (2004), 16 pp.
Johann Cigler, Some remarks and conjectures related to lattice paths in strips along the x-axis, arXiv:1501.04750 [math.CO], 2015.
Manosij Ghosh Dastidar and Michael Wallner, Bijections and congruences involving lattice paths and integer compositions, arXiv:2402.17849 [math.CO], 2024. See p. 16.
Jesús A. De Loera, Jörg Rambau, and Francisco Santos Leal, Triangulations of Point Sets [From Tom Copeland Oct 11 2011]
Satyan L. Devadoss, Combinatorial Equivalence of Real Moduli Spaces, Notices Amer. Math. Soc. 51 (2004), no. 6, 620-628.
Satyan L. Devadoss and Ronald C. Read, Cellular structures determined by polygons and trees, arXiv/0008145 [math.CO], 2000. [From Tom Copeland Nov 21 2017]
Anton Dochtermann, Face rings of cycles, associahedra, and standard Young tableaux, arXiv preprint arXiv:1503.06243 [math.CO], 2015.
Brian Drake, Ira M. Gessel, and Guoce Xin, Three Proofs and a Generalization of the Goulden-Litsyn-Shevelev Conjecture on a Sequence Arising in Algebraic Geometry, J. of Integer Sequences, Vol. 10 (2007), Article 07.3.7.
Cassandra Durell and Stefan Forcey, Level-1 Phylogenetic Networks and their Balanced Minimum Evolution Polytopes, arXiv:1905.09160 [math.CO], 2019.
P. Flajolet and M. Noy, Analytic Combinatorics of Non-crossing Configurations, Discrete Math., 204, 1999, 203-229.
Sergey Fomin and Nathan Reading, Root systems and generalized associahedra, Lecture notes for IAS/Park-City 2004, arXiv:math/0505518 [math.CO], 2005-2008. [From _Peter Bala_, Oct 28 2008]
Sergey Fomin and Andrei Zelevinsky, Cluster algebras I: Foundations, arXiv:math/0104151 [math.RT], 2001.
Sergey Fomin and Andrei Zelevinsky, Cluster algebras I: Foundations, J. Amer. Math. Soc. 15 (2002) no.2, 497-529.
Sergey Fomin and Andrei Zelevinsky, Y-systems and generalized associahedra, Ann. of Math. (2) 158 (2003), no. 3, 977-1018.
Ivan Geffner and Marc Noy, Counting Outerplanar Maps, Electronic Journal of Combinatorics 24(2) (2017), #P2.3.
Rijun Huang, Fei Teng, and Bo Feng, Permutation in the CHY-Formulation, arXiv:1801.08965 [hep-th], 2018.
Germain Kreweras, Sur les partitions non croisées d'un cycle, (French) Discrete Math. 1 (1972), no. 4, 333--350. MR0309747 (46 #8852).
Germain Kreweras, Sur les hiérarchies de segments, Cahiers Bureau Universitaire Recherche Opérationnelle, Cahier 20, Inst. Statistiques, Univ. Paris, 1973.
Germain Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)
Germain Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30.
Germain Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30. (Annotated scanned copy)
Thibault Manneville and Vincent Pilaud, Compatibility fans for graphical nested complexes, arXiv:1501.07152 [math.CO], 2015.
Sebastian Mizera, Combinatorics and topology of Kawai-Lewellen-Tye relations J. High Energy Phys. 2017, No. 8, Paper No. 97, 54 p. (2017).
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv:1403.5962 [math.CO], 2014.
Vincent Pilaud, Brick polytopes, lattice quotients, and Hopf algebras, arXiv:1505.07665 [math.CO], 2015.
Vincent Pilaud and V. Pons, Permutrees, arXiv:1606.09643 [math.CO], 2016-2017.
Ronald C. Read, On general dissections of a polygon, Aequat. Math. 18 (1978), 370-388.
Dean Rubine, Exercises for A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode, arXiv:2507.13045 [math.CO], 2025. See p. 9.
Rodica Simion, Convex Polytopes and Enumeration, Adv. in Appl. Math. 18 (1997) pp. 149-180.
Richard P. Stanley, Polygon dissections and standard Young tableaux, J. Comb. Theory, Ser. A, 76, 175-177, 1996.
Rekha R. Thomas, Lectures in Geometric Combinatorics [_Tom Copeland_, Oct 11 2011]

Cf. diagonals: A000012, A000096, A033275, A033276, A033277, A033278, A033279; A000108, A002054, A002055, A002056, A007160, A033280, A033281; row sums: A001003 (Schroeder numbers, first term omitted). See A086810 for another version.
A007160 is a diagonal. Cf. A001263.
With leading zero: A086810.
Cf. A019538 'faces' of the permutohedron.
Cf. A063007 (f-vectors type B associahedra), A080721 (f-vectors type D associahedra), A126216 (mirror image).
Cf. A248727 for a relation to f-polynomials of simplices.
Cf. A111785 (contracted partition array, unsigned; see a comment above).
Antidiagonal sums give A005043. - Jordan Tirrell, Jun 01 2017

[[Binomial(n-3, k)*Binomial(n+k-1, k)/(k+1): k in [0..(n-3)]]: n in [3..12]];  // G. C. Greubel, Nov 19 2018

T:=(n,k)->binomial(n-3,k)*binomial(n+k-1,k)/(k+1): seq(seq(T(n,k),k=0..n-3),n=3..12); # Muniru A Asiru, Nov 24 2018

t[n_, k_] = Binomial[n-3, k]*Binomial[n+k-1, k]/(k+1);
Flatten[Table[t[n, k], {n, 3, 12}, {k, 0, n-3}]][[1 ;; 52]] (* Jean-François Alcover, Jun 16 2011 *)

Q=(1+z-(1-(4*w+2+O(w^20))*z+z^2+O(z^20))^(1/2))/(2*(1+w)*z);for(n=3,12,for(m=1,n-2,print1(polcoef(polcoef(Q,n-2,z),m,w),", "))) \\ Hugo Pfoertner, Nov 19 2018

for(n=3,12, for(k=0,n-3, print1(binomial(n-3,k)*binomial(n+k-1,k)/(k+1), ", "))) \\ G. C. Greubel, Nov 19 2018

[[ binomial(n-3,k)*binomial(n+k-1,k)/(k+1) for k in (0..(n-3))] for n in (3..12)] # G. C. Greubel, Nov 19 2018

G.f. G = G(t, z) satisfies (1+t)*G^2 - z*(1-z-2*t*z)*G + t*z^4 = 0.
T(n, k) = binomial(n-3, k)*binomial(n+k-1, k)/(k+1) for n >= 3, 0 <= k <= n-3.
From Tom Copeland, Nov 03 2008: (Start)
Two g.f.s (f1 and f2) for A033282 and their inverses (x1 and x2) can be derived from the Drake and Barry references.
1. a: f1(x,t) = y = {1 - (2t+1) x - sqrt[1 - (2t+1) 2x + x^2]}/[2x (t+1)] = t x + (t + 2 t^2) x^2 + (t + 5 t^2 + 5 t^3) x^3 + ...
b: x1 = y/[t + (2t+1)y + (t+1)y^2] = y {1/[t/(t+1) + y] - 1/(1+y)} = (y/t) - (1+2t)(y/t)^2 + (1+ 3t + 3t^2)(y/t)^3 +...
2. a: f2(x,t) = y = {1 - x - sqrt[(1-x)^2 - 4xt]}/[2(t+1)] = (t/(t+1)) x + t x^2 + (t + 2 t^2) x^3 + (t + 5 t^2 + 5 t^3) x^4 + ...
b: x2 = y(t+1) [1- y(t+1)]/[t + y(t+1)] = (t+1) (y/t) - (t+1)^3 (y/t)^2 + (t+1)^4 (y/t)^3 + ...
c: y/x2(y,t) = [t/(t+1) + y] / [1- y(t+1)] = t/(t+1) + (1+t) y + (1+t)^2 y^2 + (1+t)^3 y^3 + ...
x2(y,t) can be used along with the Lagrange inversion for an o.g.f. (A133437) to generate A033282 and show that A133437 is a refinement of A033282, i.e., a refinement of the f-polynomials of the associahedra, the Stasheff polytopes.
y/x2(y,t) can be used along with the indirect Lagrange inversion (A134264) to generate A033282 and show that A134264 is a refinement of A001263, i.e., a refinement of the h-polynomials of the associahedra.
f1[x,t](t+1) gives a generator for A088617.
f1[xt,1/t](t+1) gives a generator for A060693, with inverse y/[1 + t + (2+t) y + y^2].
f1[x(t-1),1/(t-1)]t gives a generator for A001263, with inverse y/[t + (1+t) y + y^2].
The unsigned coefficients of x1(y t,t) are A074909, reverse rows of A135278. (End)
G.f.: 1/(1-x*y-(x+x*y)/(1-x*y/(1-(x+x*y)/(1-x*y/(1-(x+x*y)/(1-x*y/(1-.... (continued fraction). - Paul Barry, Feb 06 2009
Let h(t) = (1-t)^2/(1+(u-1)*(1-t)^2) = 1/(u + 2*t + 3*t^2 + 4*t^3 + ...), then a signed (n-1)-th row polynomial of A033282 is given by u^(2n-1)*(1/n!)*((h(t)*d/dt)^n) t, evaluated at t=0, with initial n=2. The power series expansion of h(t) is related to A181289 (cf. A086810). - Tom Copeland, Sep 06 2011
With a different offset, the row polynomials equal 1/(1 + x)*Integral_{0..x} R(n,t) dt, where R(n,t) = Sum_{k = 0..n} binomial(n,k)*binomial(n+k,k)*t^k are the row polynomials of A063007. - Peter Bala, Jun 23 2016
n-th row polynomial = ( LegendreP(n-1,2*x + 1) - LegendreP(n-3,2*x + 1) )/((4*n - 6)*x*(x + 1)), n >= 3. - Peter Bala, Feb 22 2017
n*T(n+1, k) = (4n-6)*T(n, k-1) + (2n-3)*T(n, k) - (n-3)*T(n-1, k) for n >= 4. - Fang Lixing, May 07 2019

Missing factor of 2 for expansions of f1 and f2 added by Tom Copeland, Apr 12 2009

A103210 a(n) = (1/n) * Sum_{i=0..n-1} C(n,i)C(n,i+1)2^i*3^(n-i), a(0)=1.

Original entry on oeis.org

1, 3, 15, 93, 645, 4791, 37275, 299865, 2474025, 20819307, 178003815, 1541918901, 13503125805, 119352115551, 1063366539315, 9539785668657, 86104685123025, 781343125570515, 7124072211203775, 65233526296899981, 599633539433039445, 5531156299278726663

Offset: 0

Ralf Stephan, Jan 27 2005

nonn

The Hankel transform of this sequence is 6^C(n+1,2). - Philippe Deléham, Oct 28 2007
The Hankel transform of the sequence starting 1, 1, 3, 15, ... is A081955. - Paul Barry, Dec 09 2008
Number of Schroeder paths from (0,0) to (0,2n) allowing two colors for the down steps (or alternatively for the rise steps). - Paul Barry, Feb 01 2009
Essentially, reversion of x*(1-2*x)/(1+x). - Paul Barry, Apr 28 2009
a(n) is also the number of infix expressions with n variables and operators +, - and * (or +, * and /) such that there are no redundant parentheses. - Vjeran Crnjak, Apr 25 2020

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Joseph Abate and Ward Whitt, Integer Sequences from Queueing Theory, J. Int. Seq. 13 (2010), 10.5.5. b_n(2).
Eyal Ackerman, Gill Barequet, Ron Y. Pinter, and Dan Romik, The number of guillotine partitions in d dimensions, Inf. Proc. Lett. (2006) Vol. 98, No. 4, 162-167.
Paul Barry, On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.
Veronica Bitonti, Bishal Deb, and Alan D. Sokal, Thron-type continued fractions (T-fractions) for some classes of increasing trees, arXiv:2412.10214 [math.CO], 2024. See p. 58.
Zhi Chen and Hao Pan, Identities involving weighted Catalan-Schroder and Motzkin Paths, arXiv:1608.02448 [math.CO], 2016, eq. (1.13), a=3, b=2.
Robert Dickau, 3D Guillotine Partitions, figures for 3D slices.
Samuele Giraudo, Operads from posets and Koszul duality, arXiv preprint arXiv:1504.04529 [math.CO], 2015.
Samuele Giraudo, Pluriassociative algebras II: The polydendriform operad and related operads, arXiv:1603.01394 [math.CO], 2016.
Luis Verde-Star, A Matrix Approach to Generalized Delannoy and Schröder Arrays, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.

Third column of array A103209.

Cf. A000108, A060693, A081955, A107841.

R:=PowerSeriesRing(Rationals(), 25); Coefficients(R!((1-x-Sqrt(x^2-10*x+1))/(4*x))); // G. C. Greubel, Feb 10 2018

A103210 := proc(n)
    if n = 0 then
        1;
    else
        add(binomial(n,i)*binomial(n,i+1)*2^i*3^(n-i),i=0..n-1)/n ;
    end if;
end proc: # R. J. Mathar, Feb 10 2015
A103210_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := 3*a[w-1] + 2*add(a[j]*a[w-j-1], j=1..w-1) od;
convert(a, list) end: A103210_list(21); # Peter Luschny, Feb 29 2016

CoefficientList[Series[(1-x-Sqrt[x^2-10*x+1])/(4*x), {x, 0, 25}], x] (* Vaclav Kotesovec, Oct 17 2012 *)
A103210[n_]:= Hypergeometric2F1[-n, n+1, 2, -2]; Table[A103210[n], {n, 0, 25}] (* Peter Luschny, Jan 07 2018 *)

my(x='x+O('x^25)); Vec((1-x-sqrt(x^2-10*x+1))/(4*x)) \\ G. C. Greubel, Feb 10 2018

[1]+[(3^n/n)*sum( binomial(n,j)*binomial(n,j+1)*(2/3)^j for j in (0..n-1)) for n in (1..25)] # G. C. Greubel, Jun 08 2020

G.f.: (1 - z - sqrt(1 -10*z +z^2))/(4*z).
a(n) = Sum_{k=0..n} C(n+k, 2k) * 2^k * C(k), C(n) given by A000108. - Paul Barry, May 21 2005
a(n) = Sum_{k=0..n} A060693(n,k)*2^(n-k). - Philippe Deléham, Apr 02 2007
a(0) = 1, a(n) = a(n-1) + 2*Sum_{k=0..n-1} a(k)*a(n-1-k). - Philippe Deléham, Oct 23 2007
a(n) = (3/2)*A107841(n) for n > 0. - Philippe Deléham, Oct 28 2007
G.f.: 1/(1-x-2*x/(1-x-2*x/(1-x-2*x/(1-.... (continued fraction). - Paul Barry, Feb 01 2009
G.f.: 1/(1-3*x-6*x^2/(1-5*x-6*x^2/(1-5*x-6*x^2/(1-... (continued fraction). - Paul Barry, Apr 28 2009
G.f.: 1/(1-3*x/(1-2*x/(1-3*x/(1-2*x/(1-3*x/(1-... (continued fraction). - Paul Barry, May 14 2009
a(n) = Hypergeometric2F1(-n,n+1;2;-2) = Sum_{k=0..n} C(n+k,k) * C(n,k) * 2^k/(k+1). - Paul Barry, Feb 08 2011
G.f.: A(x) = (1-x-(x^2-10*x+1)^(1/2))/(4*x) = 1/(G(0)-x); G(k)= 1 + x - 3*x/G(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Jan 05 2012
D-finite with recurrence: (n+1)*a(n) = 5*(2*n-1)*a(n-1) - (n-2)*a(n-2). - Vaclav Kotesovec, Oct 17 2012
a(n) ~ sqrt(12+5*sqrt(6))*(5+2*sqrt(6))^n/(4*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 17 2012
G.f. A(x) satisfies: A(x) = (1 + 2*x*A(x)^2) / (1 - x). - Ilya Gutkovskiy, Jun 30 2020

Spelling/notation corrections by Charles R Greathouse IV, Mar 18 2010

A097610 Triangle read by rows: T(n,k) is number of Motzkin paths of length n and having k horizontal steps.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 2, 0, 6, 0, 1, 0, 10, 0, 10, 0, 1, 5, 0, 30, 0, 15, 0, 1, 0, 35, 0, 70, 0, 21, 0, 1, 14, 0, 140, 0, 140, 0, 28, 0, 1, 0, 126, 0, 420, 0, 252, 0, 36, 0, 1, 42, 0, 630, 0, 1050, 0, 420, 0, 45, 0, 1, 0, 462, 0, 2310, 0, 2310, 0, 660, 0, 55, 0, 1, 132, 0, 2772, 0

Offset: 0

Emeric Deutsch, Aug 30 2004

Row sums are the Motzkin numbers (A001006). Column 0 gives the aerated Catalan numbers (A000108).
Let P_n(x) = Sum_{k=0..n} T(n,k)*x^k. P_0(x) = 1, P_1(x) = x, P_n(x) = x*P_(n-1)(x) + Sum_{j=0..n-2} P_j(x)*P_(n-2-j)(x); essentially the same as A124027. - Philippe Deléham, Oct 03 2007
G. J. Chaitin's numbers of s-expressions of size n are given by the coefficients of polynomials p(k, x) satisfying: p(k, x) = Sum_{j=2..k-1} p(j, x)*p(k-j, x). The coefficients of these polynomials also give (essentially) the triangle shown here. - Roger L. Bagula, Oct 31 2006
Exponential Riordan array [Bessel_I(1,2x)/x,x]. - Paul Barry, Mar 24 2010
Diagonal sums are the aerated large Schroeder numbers. - Paul Barry, Apr 21 2010
Non-vanishing antidiagonals are rows of A060693. - Tom Copeland, Feb 03 2016
These polynomials are related to the Gegenbauer polynomials which in turn are specializations of the Jacobi polynomials. The o.g.f. of the Gegenbauer polynomials is 1 / [1-2tx+x^2]^a. For the generating function Gb(x,h1,h2,a) = [x / (1 + h1 x + h2 x^2)]^a, the compositional inverse in x is Gbinv(x,h1,h2,a) = [(1-h1*y) - sqrt[(1-h1*y)^2-4h2*y^2]]/(2*h2*y) with y = x^(1/a). The polynomials of this entry are generated by Gbinv(x,t,1,1). The Legendre polynomials are related to the o.g.f. Gb(x,-2t,1,1/2). Cf. A121448. - Tom Copeland, Feb 07 2016
The bivariate o.g.f. in Copeland's Jan 29 2016 formulas can be related to conformal mappings of the complex plane and a solution of the dKP hierarchy. Cf. p. 24 of the Takebe et al. paper. - Tom Copeland, May 30 2018

			Triangle begins:
1;
0,  1;
1,  0,  1;
0,  3,  0,  1;
2,  0,  6,  0,  1;
0, 10,  0, 10,  0,  1;
5,  0, 30,  0, 15,  0,  1;
Row n has n+1 terms.
T(4,2)=6 because we have HHUD, HUDH, UDHH, HUHD, UHDH, UHHD, where U=(1,1), D=(1,-1) and H=(1,0).

G. J. Chaitin, Algorithmic Information Theory, Cambridge Univ. Press, 1987, page 169.

Peter J. C. Moses, Table of n, a(n) for n = 0..1080
N. Alexeev, J. Andersen, R. Penner, and P. Zograf, Enumeration of chord diagrams on many intervals and their non-orientable analogs, arXiv:1307.0967 [math.CO], 2013-2014.
M. Artioli, G. Dattoli, S. Licciardi, and S. Pagnutti, Motzkin Numbers: an Operational Point of View, arXiv:1703.07262 [math.CO], 2017, (reverse of Table 1 on p. 3).
Marilena Barnabei, Flavio Bonetti, and Niccolò Castronuovo, Motzkin and Catalan Tunnel Polynomials, J. Int. Seq., Vol. 21 (2018), Article 18.8.8.
Paul Barry, On the inversion of Riordan arrays, arXiv:2101.06713 [math.CO], 2021.
Paul Barry, On the duals of the Fibonacci and Catalan-Fibonacci polynomials and Motzkin paths, arXiv:2101.10218 [math.CO], 2021.
Tom Copeland, The Elliptic Lie Triad: Riccati and KdV Equations, Infinigens, and Elliptic Genera
Colin Defant, Postorder Preimages, arXiv preprint arXiv:1604.01723 [math.CO], 2016.
G. Ellingsrud, Elliptic curves--Basics, course notes, Fall 2014.
P. Landweber, D. Ravenel, and R. Stong, Periodic cohomology theories defined by elliptic curves
Piera Manara and Claudio Perelli Cippo, The fine structure of 4321 avoiding involutions and 321 avoiding involutions, PU. M. A. Vol. 22 (2011), 227-238. - From _N. J. A. Sloane_, Oct 13 2012
T. Takebe, Lee-Peng Teo, and A. Zabrodin, Löwner equations and dispersionless hierarchies, arXiv:math/0605161 [math.CV], p. 24, 2006.

Cf. A001006, A000108. A124027 is an essentially identical triangle.
Cf. A011973, A049310, A055151, A060693, A180874, A121448, A125181, A126120, A134264, A145271, A238390.
Cf. A001263.

G:=(1-t*z-sqrt(1-2*t*z+t^2*z^2-4*z^2))/2/z^2:
Gser:=simplify(series(G,z=0,16)): P[0]:=1:
for n from 1 to 13 do P[n]:=sort(coeff(Gser,z^n)) od:
seq(seq(coeff(t*P[n],t^k),k=1..n+1),n=0..13);
# Maple program for the triangular array:
T:=proc(n,k) if n-k mod 2 = 0 and k<=n then n!/k!/((n-k)/2)!/((n-k)/2+1)! else 0 fi end: TT:=(n,k)->T(n-1,k-1): matrix(10,10,TT);

T[n_,k_]:=If[n>=k&&EvenQ[n-k],n!/(k!((n-k)/2)!((n-k)/2+1)!),0];
Flatten[Table[T[n,k],{n,0,20},{k,0,n}]] (* Peter J. C. Moses, Apr 06 2013 *)
T[n_,k_] := If[OddQ[n - k], 0, Binomial[n, k] CatalanNumber[(n - k)/2]]; (* Peter Luschny, Jun 06 2018 *)

G.f.: [1-tz-sqrt(1-2tz+t^2*z^2-4z^2)]/(2z^2).
T(n, k) = n!/[k!((n-k)/2)!((n-k)/2-1)! ] = A055151(n, (n-k)/2) if n-k is a nonnegative even number; otherwise T(n, k) = 0.
T(n, k) = C(n, k)*C((n-k)/2)*(1+(-1)^(n-k))/2 if k <= n, 0 otherwise. - Paul Barry, May 18 2005
T(n,k) = A121448(n,k)/2^k. - Philippe Deléham, Aug 17 2006
Sum_{k=0..n} T(n,k)*2^k = A000108(n+1). - Philippe Deléham, Aug 22 2006
Sum_{k=0..n} T(n,k)*3^k = A002212(n+1). - Philippe Deléham, Oct 03 2007
G.f.: 1/(1-x*y-x^2/(1-x*y-x^2/(1-x*y-x^2/.... (continued fraction). - Paul Barry, Dec 15 2008
Sum_{k=0..n} T(n,k)*4^k = A005572(n). - Philippe Deléham, Dec 03 2009
T(n,k) = A007318(n,k)*A126120(n-k). - Philippe Deléham, Dec 12 2009
From Tom Copeland, Jan 23 2016: (Start)
E.g.f.: M(x,t) = e^(xt) AC(t) = e^(xt) I_1(2t)/t = e(xt) * e.g.f.(A126120(t)) = e^(xt) Sum_{n>=0} t^(2n)/(n!(n+1)!) = exp[t P(.,x)].
The e.g.f. of this Appell sequence of polynomials P(n,x) is e^(xt) times the e.g.f. AC(t) of the aerated Catalan numbers A126120. AC(t) = I_1(2t)/t, where I_n(x) = T_n(d/dx) I_0(x) are the modified Bessel functions of the first kind and T_n, the Chebyshev polynomials of the first kind.
P(n,x) has the lowering and raising operators L = d/dx = D and R = d/dD log{M(x,D)} = x + d/dD log{AC(D)} = x + Sum_{n>=0} c(n) D^(2n+1)/(2n+1)! with c(n) = (-1)^n A180874(n+1), i.e., L P(n,x) = n P(n-1,x) and R P(n,x) = P(n+1,x).
(P(.,x) + y)^n = P(n,x+y) = Sum_{k=0..n} binomial(n,k) P(k,x) y^(n-k) = (b.+x+y)^n, where (b.)^k = b_k = A126120(k).
Exp(b.D) e^(xt) = exp[(x+b.)t] = exp[P(.,x)t] = e^(b.t) e^(xt) = e^(xt) AC(t).
See p. 12 of the Alexeev et al. link and A055151 for a refinement.
Shifted o.g.f: G(x,t) = [1-tx-sqrt[(1-tx)^2-4x^2]] / 2x = x + t x^2 + (1+t) x^3 + ... has the compositional inverse Ginv(x,t) = x / [1 + tx + x^2] = x - t x^2 +(-1+t^2) x^3 + (2t-t^3) x^4 + (1-3t^2+t^4) x^5 + ..., a shifted o.g.f. for the signed Chebyshev polynomials of the second kind of A049310 (cf. also the Fibonacci polynomials of A011973). Then the inversion formula of A134264, involving non-crossing partitions and free probability with their multitude of interpretations (cf. A125181 also), can be used with h_0 = 1, h_1 = t, and h_2 = 1 to interpret the coefficients of the Motzkin polynomials combinatorially.
(End)
From Tom Copeland, Jan 29 2016: (Start)
Provides coefficients of the inverse of f(x) = x / [1 + h1 x + h2 x^2], a bivariate generating function of A049310 (mod signs).
finv(x) = [(1-h1*x) - sqrt[(1-h1*x)^2-4h2*x^2]]/(2*h2*x) = x + h1 x^2 + (h2 + h1^2) x^3 + (3 h1 h2 + h1^3) x^4 + ... is a bivariate o.g.f. for this entry.
The infinitesimal generator for finv(x) is g(x) d/dx with g(x) = 1 /[df(x)/dx] = x^2 / [(f(x))^2 (1 - h2 x^2)] = (1 + h1 x + h2 x^2)^2 / (1 - h2 x^2) so that [g(x)d/dx]^n/n! x evaluated at x = 0 gives the row polynomials FI(n,h1,h2) of the compositional inverse of f(x), i.e., exp[x g(u)d/du] u |_(u=0) = finv(x) = 1 / [1 -x FI(.,h1,h2)]. Cf. A145271. E.g.,
FI(0,h1,h2) = 0
FI(1,h1,h2) = 1
FI(2,h1,h2) = 1 h1
FI(3,h1,h2) = 1 h2 + 1 h1^2
FI(4,h1,h2) = 3 h2 h1 + 1 h1^3
FI(5,h1,h2) = 2 h2^2 + 6 h2 h1^2 + 1 h1^4
FI(6,h1,h2) = 10 h2^2 h1 + 10 h2 h1^3 + 1 h1^5.
And with D = d/dh1, FI(n+1, h1,h2) = MT(n,h1,h2) = (b.y + h1)^n = Sum_{k=0..n} binomial(n,k) b(k) y^k h1^(n-k) = exp[(b.y D] (h1)^n = AC(y D) (h1)^n, where b(k) = A126120(k), y = sqrt(h2), and AC(t) is defined in my Jan 23 formulas above. Equivalently, AC(y D) e^(x h1) = exp[x MT(.,h1,h2)].
The MT polynomials comprise an Appell sequence in h1 with e.g.f. e^(h1*x) AC(xy) = exp[x MT(.,h1,h2)] with lowering operator L = d/dh1 = D, i.e., L MT(n,h1,h2) = dMT(n,h1,h2)/dh1 = n MT(n-1,h1,h2) and raising operator R = h1 + dlog{AC(y L)}/dL = h1 + Sum_{n>=0} c(n) h2^(n+1) D^(2n+1)/(2n+1)! = h1 + h2 d/dh1 - h2^2 (d/dh1)^3/3! + 5 h2^3 (d/dh1)^5/5! - ... with c(n) = (-1)^n A180874(n+1) (consistent with the raising operator in the Jan 23 formulas).
The compositional inverse finv(x) is also obtained from the non-crossing partitions of A134264 (or A125181) with h_0 = 1, h_1 = h1, h_2 = h2, and h_n = 0 for all other n.
See A238390 for the umbral compositional inverse in h1 of MT(n,h1,h2) and inverse matrix.
(End)
From Tom Copeland, Feb 13 2016: (Start)
z1(x,h1,h2) = finv(x), the bivariate o.g.f. above for this entry, is the zero that vanishes for x=0 for the quadratic polynomial Q(z;z1(x,h1,h2),z2(x,h1,h2)) = (z-z1)(z-z2) = z^2 - (z1+z2) z + (z1*z2) = z^2 - e1 z + e2 = z^2 - [(1-h1*x)/(h2*x)] z + 1/h2, where e1 and e2 are the elementary symmetric polynomials for two indeterminates.
The other zero is given by z2(x,h1,h2) = (1 - h1*x)/(h2*x) - z1(x,h1,h2) = [1 - h1*x + sqrt[(1-h1*x)^2 - 4 h2*x^2]] / (2h2*x).
The two are the nontrivial zeros of the elliptic curve in Legendre normal form y^2 = z (z-z1)(z-z2), (see Landweber et al., p. 14, Ellingsrud, and A121448), and the zeros for the Riccati equation z' = (z - z1)(z - z2), associated to soliton solutions of the KdV equation (see Copeland link).
(End)
Comparing the shifted o.g.f. S(x) = x / (1 - h_1 x + h_2 x^2) for the bivariate Chebyshev polynomials S_n(h_1,h_2) of A049310 with the shifted o.g.f. H(x) = x / ((1 - a x)(1 - b x)) for the complete homogeneous symmetric polynomials H_n(a,b) = (a^(n+1)-b^(n+1)) / (a - b) shows that S_n(h_1,h_2) = H_n(a,b) for h_1 = a + b and h_2 = ab and, conversely, a = (h_1 + sqrt(h_1^2 - 4 h_2)) / 2 and b = (h_1 - sqrt(h_1^2 - 4 h_2)) / 2. The compositional inverse about the origin of S(x) gives a bivariate o.g.f. for signed Motzkin polynomials M_n(h_1,h_2) of this entry, and that of H(x) gives one for signed Narayana polynomials N_n(a,b) of A001263, thereby relating the bivariate Motzkin and Narayana polynomials by the indeterminate transformations. E.g., M_2(h_1,h_2) = h_2 + h_1^2 = ab + (a + b)^2 = a^2 + 3 ab + b^2 = N_2(a,b). - Tom Copeland, Jan 27 2024

cp's OEIS Frontend

A006318 Large Schröder numbers (or large Schroeder numbers, or big Schroeder numbers).

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A090181 Triangle of Narayana (A001263) with 0 <= k <= n, read by rows.

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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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A033282 Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.

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A103210 a(n) = (1/n) * Sum_{i=0..n-1} C(n,i)*C(n,i+1)*2^i*3^(n-i), a(0)=1.

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Magma

A103210 a(n) = (1/n) * Sum_{i=0..n-1} C(n,i)C(n,i+1)2^i*3^(n-i), a(0)=1.

A103211 a(n) = (1/n) * Sum_{i=0..n-1} C(n,i)C(n,i+1)3^i*4^(n-i), a(0)=1.

A082298 Expansion of (1-3x-sqrt(9x^2-10x+1))/(2x).

A007004 a(n) = (3n)! / ((n+1)(n!)^3).