This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A006318 M1659 #793 Jul 03 2025 01:08:57
%S A006318 1,2,6,22,90,394,1806,8558,41586,206098,1037718,5293446,27297738,
%T A006318 142078746,745387038,3937603038,20927156706,111818026018,600318853926,
%U A006318 3236724317174,17518619320890,95149655201962,518431875418926,2832923350929742,15521467648875090
%N A006318 Large Schröder numbers (or large Schroeder numbers, or big Schroeder numbers).
%C A006318 For the little Schröder numbers (or little Schroeder numbers, or small Schroeder numbers) see A001003.
%C A006318 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
%C A006318 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
%C A006318 Twice A001003 (except for the first term).
%C A006318 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
%C A006318 a(n) is the number of separable permutations, i.e., permutations avoiding 2413 and 3142 (see Shapiro and Stephens). - _Vincent Vatter_, Aug 16 2006
%C A006318 _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
%C A006318 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
%C A006318 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
%C A006318 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
%C A006318 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
%C A006318 Triangle A144156 has row sums equal to A006318 with left border A001003. - _Gary W. Adamson_, Sep 12 2008
%C A006318 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
%C A006318 Sum_{n >= 0} a(n)/10^n - 1 = (9 - sqrt(41))/2. - _Mark Dols_, Jun 22 2010
%C A006318 1/sqrt(41) = Sum_{n >= 0} Delannoy number(n)/10^n. - _Mark Dols_, Jun 22 2010
%C A006318 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
%C A006318 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
%C A006318 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
%C A006318 From _Jon Perry_, May 24 2013: (Start)
%C A006318 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.
%C A006318 Triangle begins:
%C A006318 1;
%C A006318 1, 2;
%C A006318 1, 4, 6;
%C A006318 1, 6, 16, 22;
%C A006318 1, 8, 30, 68, 90;
%C A006318 ... (End)
%C A006318 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
%C A006318 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
%C A006318 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
%C A006318 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
%C A006318 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
%C A006318 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
%C A006318 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
%C A006318 Named after the German mathematician Ernst Schröder (1841-1902). - _Amiram Eldar_, Apr 15 2021
%C A006318 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
%C A006318 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
%C A006318 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
%C A006318 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
%C A006318 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
%D A006318 D. Andrica and E. J. Ionascu, On the number of polynomials with coefficients in [n], An. St. Univ. Ovidius Constanta, 2013, to appear.
%D A006318 Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
%D A006318 Paul Barry, Riordan-Bernstein Polynomials, Hankel Transforms and Somos Sequences, Journal of Integer Sequences, Vol. 15 2012, #12.8.2.
%D A006318 Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.
%D A006318 Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
%D A006318 O. Bodini, A. Genitrini, F. Peschanski, and N.Rolin, Associativity for binary parallel processes, CALDAM 2015.
%D A006318 Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, pages 24, 618.
%D A006318 S. Brlek, E. Duchi, E. Pergola, and S. Rinaldi, On the equivalence problem for succession rules, Discr. Math., 298 (2005), 142-154.
%D A006318 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.
%D A006318 William Y. C. Chen and Carol J. Wang, Noncrossing Linked Partitions and Large (3, 2)-Motzkin Paths, Discrete Math., 312 (2012), 1918-1922.
%D A006318 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 81, #21, (4), q_n.
%D A006318 D. E. Davenport, L. W. Shapiro, and L. C. Woodson, The Double Riordan Group, The Electronic Journal of Combinatorics, 18(2) (2012), #P33.
%D A006318 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.
%D A006318 E. Deutsch, A bijective proof of an equation linking the Schroeder numbers, large and small, Discrete Math., 241 (2001), 235-240.
%D A006318 C. Domb and A. J. Barrett, Enumeration of ladder graphs, Discrete Math. 9 (1974), 341-358.
%D A006318 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
%D A006318 M. Dziemianczuk, Generalizing Delannoy numbers via counting weighted lattice paths, INTEGERS, 13 (2013), #A54.
%D A006318 Egge, Eric S., Restricted signed permutations counted by the Schröder numbers. Discrete Math. 306 (2006), 552-563. [Many applications of these numbers.]
%D A006318 S. Getu et al., How to guess a generating function, SIAM J. Discrete Math., 5 (1992), 497-499.
%D A006318 S. Gire, Arbres, permutations a motifs exclus et cartes planaire: quelques problemes algorithmiques et combinatoires, Ph.D. Thesis, Universite Bordeaux I, 1993.
%D A006318 N. S. S. Gu, N. Y. Li, and T. Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.
%D A006318 Guruswami, Venkatesan, Enumerative aspects of certain subclasses of perfect graphs. Discrete Math. 205 (1999), 97-117.
%D A006318 Silvia Heubach and Toufik Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.
%D A006318 D. E. Knuth, The Art of Computer Programming, Vol. 1, Section 2.2.1, Problem 11.
%D A006318 D. Kremer, Permutations with forbidden subsequences and a generalized Schröder number, Discrete Math. 218 (2000) 121-130.
%D A006318 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.
%D A006318 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
%D A006318 L. Moser and W. Zayachkowski, Lattice paths with diagonal steps, Scripta Math., 26 (1961), 223-229.
%D A006318 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.
%D A006318 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A006318 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see page 178 and also Problems 6.39 and 6.40.
%D A006318 Lin Yang and S.-L. Yang, The parametric Pascal rhombus. Fib. Q., 57:4 (2019), 337-346.
%D A006318 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.
%H A006318 Fung Lam, <a href="/A006318/b006318.txt">Table of n, a(n) for n = 0..2000</a> (terms 0..100 by T. D. Noe)
%H A006318 J. Abate and W. Whitt, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Whitt/whitt2.html">Integer Sequences from Queueing Theory</a>, J. Int. Seq. 13 (2010), 10.5.5, Corollary 8.
%H A006318 M. Aigner, <a href="https://doi.org/10.1016/j.disc.2007.06.012">Enumeration via ballot numbers</a>, Discrete Math., 308 (2008), 2544-2563.
%H A006318 Andrei Asinowski and Cyril Banderier, <a href="https://arxiv.org/abs/2401.05558">From geometry to generating functions: rectangulations and permutations</a>, arXiv:2401.05558 [cs.DM], 2024. See page 2.
%H A006318 Andrei Asinowski, G. Barequet, M. Bousquet-Mélou, T. Mansour, and R. Pinter, <a href="https://arxiv.org/abs/1011.1889">Orders induced by segments in floorplans and (2-14-3,3-41-2)-avoiding permutations</a>, arXiv:1011.1889 [math.CO], 2010-2012.
%H A006318 M. D. Atkinson and T. Stitt, <a href="http://www.cs.otago.ac.nz/staffpriv/mike/Papers/WreathProduct/Wreathpaper.pdf">Restricted permutations and the wreath product</a>, Preprint, 2002.
%H A006318 M. D. Atkinson and T. Stitt, <a href="https://doi.org/10.1016/S0012-365X(02)00443-0">Restricted permutations and the wreath product</a>, Discrete Math., 259 (2002), 19-36.
%H A006318 Yu Hin Au, <a href="https://arxiv.org/abs/1912.00555">Some Properties and Combinatorial Implications of Weighted Small Schröder Numbers</a>, arXiv:1912.00555 [math.CO], 2019.
%H A006318 Jean-Christophe Aval and F. Bergeron, <a href="https://arxiv.org/abs/1603.09487">Rectangular Schroder Parking Functions Combinatorics</a>, arXiv:1603.09487 [math.CO], 2016.
%H A006318 Axel Bacher, <a href="https://arxiv.org/abs/1802.06030">Improving the Florentine algorithms: recovering algorithms for Motzkin and Schröder paths</a>, arXiv:1802.06030 [cs.DS], 2018.
%H A006318 C. Banderier and D. Merlini, <a href="http://algo.inria.fr/banderier/Papers/infjumps.ps">Lattice paths with an infinite set of jumps</a>, FPSAC02, Melbourne, 2002.
%H A006318 Kayleigh Bangs, Skye Binegar, Young Kim, Kyle Ormsby, Angélica M. Osorno, David Tamas-Parris, and Livia Xu, <a href="https://arxiv.org/abs/1907.00933">Biased permutative equivariant categories</a>, arXiv:1907.00933 [math.AT], 2019.
%H A006318 E. Barcucci, A. Del Lungo, E. Pergola, and R. Pinzani, <a href="https://www.emis.de/journals/DMTCS/pdfpapers/dm040103.pdf">Permutations avoiding an increasing number of length-increasing forbidden subsequences</a>, Discrete Mathematics and Theoretical Computer Science, 4 (2000), 31-44.
%H A006318 E. Barcucci, A. Del Lungo, E. Pergola, and R. Pinzani, <a href="https://doi.org/10.1016/S0012-365X(00)00359-9">Some permutations with forbidden subsequences and their inversion number</a>, Discrete Math. 234(1-3) (2001), 1-15.
%H A006318 E. Barcucci, E. Pergola, R. Pinzani, and S. Rinaldi, <a href="http://www.mat.univie.ac.at/~slc/wpapers/s46rinaldi.html">ECO method and hill-free generalized Motzkin paths</a>, Séminaire Lotharingien de Combinatoire, B46b (2001), 14 pp.
%H A006318 J.-L. Baril, C. Khalil, and V. Vajnovszki, <a href="https://arxiv.org/abs/2004.01812">Catalan and Schröder permutations sortable by two restricted stacks</a>, arXiv:2004.01812 [cs.DM], 2020.
%H A006318 Marilena Barnabei, Flavio Bonetti, and Niccolò Castronuovo, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Barnabei/barnabei5.html">Motzkin and Catalan Tunnel Polynomials</a>, J. Int. Seq., Vol. 21 (2018), Article 18.8.8.
%H A006318 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Barry3/barry93.html">Continued fractions and transformations of integer sequences</a>, JIS 12 (2009) 09.7.6
%H A006318 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Barry1/barry95r.html">Generalized Catalan Numbers, Hankel Transforms and Somos-4 Sequences </a>, J. Int. Seq. 13 (2010) #10.7.2.
%H A006318 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Barry4/barry142.html">On a Generalization of the Narayana Triangle</a>, J. Int. Seq. 14 (2011), Article 11.4.5.
%H A006318 Paul Barry, <a href="https://arxiv.org/abs/1311.2292">Laurent Biorthogonal Polynomials and Riordan Arrays</a>, arXiv preprint arXiv:1311.2292 [math.CA], 2013.
%H A006318 Paul Barry, <a href="https://arxiv.org/abs/1802.03443">On a transformation of Riordan moment sequences</a>, arXiv:1802.03443 [math.CO], 2018.
%H A006318 Paul Barry, <a href="https://www.emis.de/journals/JIS/VOL22/Barry3/barry422.html">Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles</a>, J. Int. Seq., 22 (2019), Article 19.5.8.
%H A006318 Paul Barry, <a href="https://arxiv.org/abs/1910.00875">Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials</a>, arXiv:1910.00875 [math.CO], 2019.
%H A006318 Paul Barry, <a href="https://arxiv.org/abs/1912.01126">Riordan arrays, the A-matrix, and Somos 4 sequences</a>, arXiv:1912.01126 [math.CO], 2019.
%H A006318 Paul Barry, <a href="https://arxiv.org/abs/2504.09719">Notes on Riordan arrays and lattice paths</a>, arXiv:2504.09719 [math.CO], 2025. See pp. 25, 29.
%H A006318 Paul Barry and Aoife Hennessy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Barry2/barry126.html">A Note on Narayana Triangles and Related Polynomials, Riordan Arrays, and MIMO Capacity Calculations</a>, J. Int. Seq. 14 (2011), Article 11.3.8.
%H A006318 Paul Barry and Nikolaos Pantelidis,<a href="https://doi.org/10.1007/s10801-020-00993-w">On pseudo-involutions, involutions and quasi-involutions in the group of almost Riordan arrays</a>, J Algebr Comb 54, 399-423 (2021). (appeared in its aerated form,i.e. 1,0,2,0,6,0,...)
%H A006318 Christian Bean, Émile Nadeau, and Henning Ulfarsson, <a href="https://arxiv.org/abs/1912.07503">Enumeration of Permutation Classes and Weighted Labelled Independent Sets</a>, arXiv:1912.07503 [math.CO], 2019.
%H A006318 Arkady Berenstein, Vladimir Retakh, Christophe Reutenauer, and Doron Zeilberger, <a href="https://arxiv.org/abs/1206.4225">The Reciprocal of Sum_{n >= 0} a^n b^n for non-commuting a and b, Catalan numbers and non-commutative quadratic equations</a>, arXiv preprint arXiv:1206.4225 [math.CO], 2012. - From _N. J. A. Sloane_, Nov 28 2012
%H A006318 F. R. Bernhart and N. J. A. Sloane, <a href="/A006343/a006343.pdf">Emails, April-May 1994</a>.
%H A006318 J. Bloom and A. Burstein, <a href="https://arxiv.org/abs/1410.0230">Egge triples and unbalanced Wilf-equivalence</a>, arXiv:1410.0230 [math.CO], 2014.
%H A006318 O. Bodini, A. Genitrini, and F. Peschanski, <a href="http://www-apr.lip6.fr/~genitrini/publi/fsttcs13_genitrini.pdf">The Combinatorics of Non-determinism</a>, 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.
%H A006318 Miklós Bóna, Cheyne Homberger, Jay Pantone, and Vince Vatter, <a href="https://arxiv.org/abs/1310.7003">Pattern-avoiding involutions: exact and asymptotic enumeration</a>, arxiv:1310.7003 [math.CO], 2013-2014.
%H A006318 M. Bremner and S. Madariaga, <a href="https://arxiv.org/abs/1408.3069">Lie and Jordan products in interchange algebras</a>, arXiv:1408.3069 [math.RA], 2014-2015.
%H A006318 M. Bremner and S. Madariaga, <a href="https://arxiv.org/abs/1405.2889">Permutation of elements in double semigroups</a>, arXiv:1405.2889 [math.RA], 2014-2015.
%H A006318 R. Brignall, S. Huczynska, and V. Vatter, <a href="https://arXiv.org/abs/math.CO/0608391">Simple permutations and algebraic generating functions</a>, arXiv:math/0608391 [math.CO], 2006.
%H A006318 Marie-Louise Bruner and Martin Lackner, <a href="https://arxiv.org/abs/1505.05852">On the Likelihood of Single-Peaked Preferences</a>, arXiv:1505.05852 [cs.GT], 2015.
%H A006318 Alexander Burstein, Sergi Elizalde, and Toufik Mansour, <a href="https://arXiv.org/abs/math.CO/0610234">Restricted Dumont permutations, Dyck paths and noncrossing partitions</a>, arXiv:math/0610234 [math.CO], 2006. See Theorem 3.5.
%H A006318 Alexander Burstein and J. Pantone, <a href="https://arxiv.org/abs/1402.3842">Two examples of unbalanced Wilf-equivalence</a>, arXiv:1402.3842 [math.CO], 2014.
%H A006318 Alexander Burstein and Louis W. Shapiro, <a href="https://arxiv.org/abs/2112.11595">Pseudo-involutions in the Riordan group</a>, arXiv:2112.11595 [math.CO], 2021.
%H A006318 David Callan, <a href="https://arxiv.org/abs/1210.6455">An application of a bijection of Mansour, Deng, and Du</a>, arXiv:1210.6455 [math.CO], 2012.
%H A006318 David Callan, <a href="https://arxiv.org/abs/1602.05571">A note on a bijection for Schröder permutations</a>, arXiv:1602.05571 [math.CO], 2016.
%H A006318 David Callan and Toufik Mansour, <a href="https://arxiv.org/abs/1602.05182">Five subsets of permutations enumerated as weak sorting permutations</a>, arXiv:1602.05182 [math.CO], 2016.
%H A006318 Hui-Qin Cao and Hao Pan, <a href="https://arxiv.org/abs/1512.06310">A Stern-type congruence for the Schröder numbers</a>, arXiv:1512.06310 [math.NT], 2015.
%H A006318 Jean Cardinal, Vera Sacristán, and Rodrigo I. Silveira, <a href="https://arxiv.org/abs/1712.07919">A Note on Flips in Diagonal Rectangulations</a>, arXiv:1712.07919 [math.CO], 2017.
%H A006318 F. Chapoton, F. Hivert, and J.-C. Novelli, <a href="https://arxiv.org/abs/1307.0092">A set-operad of formal fractions and dendriform-like sub-operads</a>, arXiv:1307.0092 [math.CO], 2013.
%H A006318 F. Chapoton and S. Giraudo, <a href="https://arxiv.org/abs/1310.4521">Enveloping operads and bicoloured noncrossing configurations</a>, arXiv:1310.4521 [math.CO], 2013.
%H A006318 W. Y. C. Chen, L. H. Liu, and C. J. Wang, <a href="https://arxiv.org/abs/1305.5357">Linked Partitions and Permutation Tableaux</a>, arXiv:1305.5357 [math.CO], 2013.
%H A006318 Z. Chen and H. Pan, <a href="https://arxiv.org/abs/1608.02448">Identities involving weighted Catalan-Schroder and Motzkin Paths</a>, arXiv:1608.02448 (2016), eq. (1.13), a=2, b=1.
%H A006318 Shane Chern, <a href="https://arxiv.org/abs/2006.04318">On 0012-avoiding inversion sequences and a Conjecture of Lin and Ma</a>, arXiv:2006.04318 [math.CO], 2020.
%H A006318 J. Cigler, <a href="http://homepage.univie.ac.at/johann.cigler/preprints/hankel.pdf">Hankel determinants of some polynomial sequences</a>, 2012.
%H A006318 Johann Cigler, Christian Krattenthaler, <a href="https://arxiv.org/abs/2003.01676">Hankel determinants of linear combinations of moments of orthogonal polynomials</a>, arXiv:2003.01676 [math.CO], 2020.
%H A006318 M. Ciucu, <a href="https://www.researchgate.net/publication/225203580_Perfect_Matchings_of_Cellular_Graphs">Perfect matchings of cellular graphs</a>, J. Algebraic Combin., 5 (1996) 87-103.
%H A006318 CombOS - Combinatorial Object Server, <a href="http://combos.org/rect">Generate slicing floorplans</a>
%H A006318 Sylvie Corteel, Megan A. Martinez, Carla D. Savage, and Michael Weselcouch, <a href="https://arxiv.org/abs/1510.05434">Patterns in Inversion Sequences I</a>, arXiv:1510.05434 [math.CO], 2015
%H A006318 S. Crowley, <a href="http://vixra.org/abs/1202.0079">Mellin and Laplace Integral Transforms Related to the Harmonic Sawtooth Map and a Diversion Into The Theory Of Fractal Strings</a>, Number Theory, viXra:1202.0079, 2012.
%H A006318 R. De Castro, A. L. Ramírez, and J. L. Ramírez, <a href="https://arxiv.org/abs/1310.2339">Applications in Enumerative Combinatorics of Infinite Weighted Automata and Graphs</a>, arXiv:1310.2449 [math.PR], 2013.
%H A006318 Colin Defant, <a href="https://arxiv.org/abs/1809.03123">Stack-sorting preimages of permutation classes</a>, arXiv:1809.03123 [math.CO], 2018.
%H A006318 Colin Defant, <a href="https://arxiv.org/abs/2004.11367">Troupes, Cumulants, and Stack-Sorting</a>, arXiv:2004.11367 [math.CO], 2020.
%H A006318 Phan Thuan Do, Thi Thu Huong Tran, and Vincent Vajnovszki, <a href="https://arxiv.org/abs/1809.00742">Exhaustive generation for permutations avoiding a (colored) regular sets of patterns</a>, arXiv:1809.00742 [cs.DM], 2018.
%H A006318 B. Drake, <a href="http://people.brandeis.edu/~gessel/homepage/students/drakethesis.pdf">An inversion theorem for labeled trees and some limits of areas under lattice paths (Example 1.6.7)</a>, A dissertation presented to the Faculty of the Graduate School of Arts and Sciences of Brandeis University.
%H A006318 D. Drake, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Drake/drake.html">Bijections from Weighted Dyck Paths to Schröder Paths</a>, J. Int. Seq. 13 (2010), #10.9.2.
%H A006318 Rosena R. X. Du, Xiaojie Fan, and Yue Zhao, <a href="https://arxiv.org/abs/1803.01590">Enumeration on row-increasing tableaux of shape 2 X n</a>, arXiv:1803.01590 [math.CO], 2018.
%H A006318 M. Dziemianczuk, <a href="https://arxiv.org/abs/1410.5747">On Directed Lattice Paths With Additional Vertical Steps</a>, arXiv preprint arXiv:1410.5747 [math.CO], 2014.
%H A006318 James East and Nicholas Ham, <a href="https://arxiv.org/abs/1811.05735">Lattice paths and submonoids of Z^2</a>, arXiv:1811.05735 [math.CO], 2018.
%H A006318 Ömer Eğecioğlu, Collier Gaiser, and Mei Yin, <a href="https://arxiv.org/abs/2309.15964">Enumerating pattern-avoiding permutations by leading terms</a>, arXiv:2309.15964 [math.CO], 2023.
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%H A006318 Luca Ferrari and Emanuele Munarini, <a href="https://arxiv.org/abs/1203.6792">Enumeration of edges in some lattices of paths</a>, arXiv:1203.6792 [math.CO], 2012 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Ferrari/ferrari.html">J. Int. Seq. 17 (2014), #14.1.5</a>
%H A006318 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; see p. 474.
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%H A006318 Christian Gaetz and Yibo Gao, <a href="https://arxiv.org/abs/1905.09331">Separable elements in Weyl groups</a>, arXiv:1905.09331 [math.CO], 2019.
%H A006318 Alice L. L. Gao and Sergey Kitaev, <a href="https://arxiv.org/abs/1903.08946">On partially ordered patterns of length 4 and 5 in permutations</a>, arXiv:1903.08946 [math.CO], 2019.
%H A006318 Alice L. L. Gao and Sergey Kitaev, <a href="https://doi.org/10.37236/8605">On partially ordered patterns of length 4 and 5 in permutations</a>, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
%H A006318 Olivier Gérard, <a href="/A006318/a006318.pdf">Illustration of initial terms</a>
%H A006318 Étienne Ghys, <a href="http://images-archive.math.cnrs.fr/Quand-beaucoup-de-courbes-se.html">Quand beaucoup de courbes se rencontrent</a> — Images des Mathématiques, CNRS, 2009.
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%H A006318 Étienne Ghys, <a href="https://arxiv.org/abs/1612.06373">A Singular Mathematical Promenade</a>, arXiv:1612.06373, 2016.
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%H A006318 E. Pergola and R. A. Sulanke, <a href="https://cs.uwaterloo.ca/journals/JIS/PergolaSulanke/">Schröder Triangles, Paths and Parallelogram Polyominoes</a>, J. Integer Sequences, 1 (1998), #98.1.7.
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%H A006318 L. W. Shapiro and N. J. A. Sloane, <a href="/A006318/a006318_1.pdf">Correspondence, 1976</a>.
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%H A006318 Paul Tarau, <a href="http://www.cse.unt.edu/~tarau/research/2015/dbt.pdf">On Type-directed Generation of Lambda Terms</a>, preprint, 2015.
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%H A006318 Yi Wang and Bao-Xuan Zhu, <a href="http://arxiv.org/abs/1303.5595">Proofs of some conjectures on monotonicity of number-theoretic and combinatorial sequences</a>, arXiv:1303.5595 [math.CO], 2013.
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%H A006318 <a href="/index/Cor#core">Index entries for "core" sequences</a>
%F A006318 G.f.: (1 - x - (1 - 6*x + x^2)^(1/2))/(2*x).
%F A006318 a(n) = 2*hypergeom([-n+1, n+2], [2], -1). - _Vladeta Jovovic_, Apr 24 2003
%F A006318 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
%F A006318 The g.f. satisfies (1 - x)*A(x) - x*A(x)^2 = 1. - _Ralf Stephan_, Jun 30 2003
%F A006318 For the asymptotic behavior, see A001003 (remembering that A006318 = 2*A001003). - _N. J. A. Sloane_, Apr 10 2011
%F A006318 From _Philippe Deléham_, Nov 28 2003: (Start)
%F A006318 Row sums of A088617 and A060693.
%F A006318 a(n) = Sum_{k = 0..n} C(n+k, n)*C(n, k)/(k+1). (End)
%F A006318 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
%F A006318 a(n) = Sum_{k = 0..n} A000108(k)*binomial(n+k, n-k). - _Benoit Cloitre_, May 09 2004
%F A006318 a(n) = Sum_{k = 0..n} A011117(n, k). - _Philippe Deléham_, Jul 10 2004
%F A006318 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
%F A006318 From _Abdullahi Umar_, Oct 11 2008: (Start)
%F A006318 A123164(n+1) - A123164(n) = (2*n+1)*a(n) (n >= 0).
%F A006318 and 2*A123164(n) = (n+1)*a(n) - (n-1)*a(n-1) (n > 0). (End)
%F A006318 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
%F A006318 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
%F A006318 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
%F A006318 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
%F A006318 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
%F A006318 Logarithmic derivative yields A002003. - _Paul D. Hanna_, Oct 25 2010
%F A006318 a(n) = the upper left term in M^(n+1), M = the production matrix:
%F A006318 1, 1, 0, 0, 0, 0, ...
%F A006318 1, 1, 1, 0, 0, 0, ...
%F A006318 2, 2, 1, 1, 0, 0, ...
%F A006318 4, 4, 2, 1, 1, 0, ...
%F A006318 8, 8, 8, 2, 1, 1, ...
%F A006318 ... - _Gary W. Adamson_, Jul 08 2011
%F A006318 a(n) is the sum of top row terms in Q^n, Q = an infinite square production matrix as follows:
%F A006318 1, 1, 0, 0, 0, 0, ...
%F A006318 1, 1, 2, 0, 0, 0, ...
%F A006318 1, 1, 1, 2, 0, 0, ...
%F A006318 1, 1, 1, 1, 2, 0, ...
%F A006318 1, 1, 1, 1, 1, 2, ...
%F A006318 ... - _Gary W. Adamson_, Aug 23 2011
%F A006318 From _Tom Copeland_, Sep 21 2011: (Start)
%F A006318 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.
%F A006318 Consequently, with H(x) = 1/ (dG(x)/dx) = (2 + 3*x + x^2)^2 / (2 - x^2),
%F A006318 a(n) = (1/n!)*[(H(x)*d/dx)^n] x evaluated at x = 0, i.e.,
%F A006318 F(x) = exp[x*H(u)*d/du] u, evaluated at u = 0. Also, dF(x)/dx = H(F(x)). (End)
%F A006318 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
%F A006318 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
%F A006318 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
%F A006318 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
%F A006318 a(n+1) = a(n) + Sum_{k=0..n} a(k)*(n-k). - _Reinhard Zumkeller_, Nov 13 2012
%F A006318 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
%F A006318 a(-1-n) = a(n). - _Michael Somos_, Apr 03 2013
%F A006318 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
%F A006318 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
%F A006318 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
%F A006318 a(n) = hypergeom([-n, n+1], [2], -1). - _Peter Luschny_, Mar 23 2015
%F A006318 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
%F A006318 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
%F A006318 From _Peter Bala_, May 13 2024: (Start)
%F A006318 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.
%F A006318 a(n) = Integral_{x = 0..1} Legendre_P(n, 2*x+1) dx. (End)
%F A006318 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
%e A006318 a(3) = 22 since the top row of Q^n = (6, 6, 6, 4, 0, 0, 0, ...); where 22 = (6 + 6 + 6 + 4).
%e A006318 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 + ...
%p A006318 Order := 24: solve(series((y-y^2)/(1+y),y)=x,y); # then A(x)=y(x)/x
%p A006318 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
%p A006318 A006318_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
%p A006318 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
%p A006318 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
%p A006318 seq(simplify(hypergeom([-n,n+1],[2],-1)), n=0..100); # _Robert Israel_, Mar 23 2015
%t A006318 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]
%t A006318 InverseSeries[Series[(y - y^2)/(1 + y), {y, 0, 24}], x] (* then A(x) = y(x)/x *) (* _Len Smiley_, Apr 11 2000 *)
%t A006318 CoefficientList[Series[(1 - x - (1 - 6x + x^2)^(1/2))/(2x), {x, 0, 30}], x] (* _Harvey P. Dale_, May 01 2011 *)
%t A006318 a[ n_] := 2 Hypergeometric2F1[ -n + 1, n + 2, 2, -1]; (* _Michael Somos_, Apr 03 2013 *)
%t A006318 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 *)
%t A006318 Table[-(GegenbauerC[n+1, -1/2, 3] + KroneckerDelta[n])/2, {n, 0, 30}] (* _Vladimir Reshetnikov_, Nov 12 2016 *)
%t A006318 CoefficientList[Nest[1+x(#+#^2)&, 1+O[x], 20], x] (* _Oliver Seipel_, Dec 21 2024 *)
%o A006318 (PARI) {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 */
%o A006318 (PARI) {a(n) = if( n<1, 1, sum( k=0, n, 2^k * binomial( n, k) * binomial( n, k-1)) / n)};
%o A006318 (Sage) # Generalized algorithm of L. Seidel
%o A006318 def A006318_list(n) :
%o A006318 D = [0]*(n+1); D[1] = 1
%o A006318 b = True; h = 1; R = []
%o A006318 for i in range(2*n) :
%o A006318 if b :
%o A006318 for k in range(h,0,-1) : D[k] += D[k-1]
%o A006318 h += 1;
%o A006318 else :
%o A006318 for k in range(1,h, 1) : D[k] += D[k-1]
%o A006318 R.append(D[h-1]);
%o A006318 b = not b
%o A006318 return R
%o A006318 A006318_list(23) # _Peter Luschny_, Jun 02 2012
%o A006318 (Haskell)
%o A006318 a006318 n = a004148_list !! n
%o A006318 a006318_list = 1 : f [1] where
%o A006318 f xs = y : f (y : xs) where
%o A006318 y = head xs + sum (zipWith (*) xs $ reverse xs)
%o A006318 -- _Reinhard Zumkeller_, Nov 13 2012
%o A006318 (Python)
%o A006318 from gmpy2 import divexact
%o A006318 A006318 = [1, 2]
%o A006318 for n in range(3,10**3):
%o A006318 A006318.append(int(divexact(A006318[-1]*(6*n-9)-(n-3)*A006318[-2],n)))
%o A006318 # _Chai Wah Wu_, Sep 01 2014
%o A006318 (GAP) 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
%Y A006318 Apart from leading term, twice A001003 (the small Schroeder numbers). Cf. A025240.
%Y A006318 Sequences A085403, A086456, A103137, A112478 are essentially the same sequence.
%Y A006318 Main diagonal of A033877.
%Y A006318 Cf. A002003, A004148, A088617, A060693, A144156.
%Y A006318 Row sums of A104219. Bisections give A138462, A138463.
%Y A006318 Row sums of A175124.
%Y A006318 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
%K A006318 nonn,easy,core,nice
%O A006318 0,2
%A A006318 _N. J. A. Sloane_
%E A006318 Edited by _Charles R Greathouse IV_, Apr 20 2010