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 A001006 Motzkin numbers: number of ways of drawing any number of nonintersecting chords joining n (labeled) points on a circle. (Formerly M1184 N0456) 510
 1, 1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, 5798, 15511, 41835, 113634, 310572, 853467, 2356779, 6536382, 18199284, 50852019, 142547559, 400763223, 1129760415, 3192727797, 9043402501, 25669818476, 73007772802, 208023278209, 593742784829 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Number of 4321-, (3412,2413)-, (3412,3142)- and 3412-avoiding involutions in S_n. Number of sequences of length n-1 consisting of positive integers such that the opening and ending elements are 1 or 2 and the absolute difference between any 2 consecutive elements is 0 or 1. - Jon Perry, Sep 04 2003 Also number of Motzkin n-paths: paths from (0,0) to (n,0) in an n X n grid using only steps U = (1,1), F = (1,0) and D = (1,-1). - David Callan, Jul 15 2004 Number of Dyck n-paths with no UUU. (Given such a Dyck n-path, change each UUD to U, then change each remaining UD to F. This is a bijection to Motzkin n-paths. Example with n=5: U U D U D U U D D D -> U F U D D.) - David Callan, Jul 15 2004 Number of Dyck (n+1)-paths with no UDU. (Given such a Dyck (n+1)-path, mark each U that is followed by a D and each D that is not followed by a U. Then change each unmarked U whose matching D is marked to an F. Lastly, delete all the marked steps. This is a bijection to Motzkin n-paths. Example with n=6 and marked steps in small type: U U u d D U U u d d d D u d -> U U u d D F F u d d d D u d -> U U D F F D.) - David Callan, Jul 15 2004 a(n) is the number of strings of length 2n+2 from the following recursively defined set: L contains the empty string and, for any strings a and b in L, we also find (ab) in L. The first few elements of L are e, (), (()), ((())), (()()), (((()))), ((()())), ((())()), (()(())) and so on. This proves that a(n) is less than or equal to C(n+1), the (n+1)-th Catalan number. - Saul Schleimer (saulsch(AT)math.rutgers.edu), Feb 23 2006 [corrected by Sergey Kirgizov, Mar 05 2020] a(n) = number of Dyck n-paths all of whose valleys have even x-coordinate (when path starts at origin). For example, T(4,2)=3 counts UDUDUUDD, UDUUDDUD, UUDDUDUD. Given such a path, split it into n subpaths of length 2 and transform UU->U, DD->D, UD->F (there will be no DUs for that would entail a valley with odd x-coordinate). This is a bijection to Motzkin n-paths. - David Callan, Jun 07 2006 Also the number of standard Young tableaux of height <= 3. - Mike Zabrocki, Mar 24 2007 a(n) is the number of RNA shapes of size 2n+2. RNA Shapes are essentially Dyck words without "directly nested" motifs of the form A[[B]]C, for A, B and C Dyck words. The first RNA Shapes are []; [][]; [][][], [[][]]; [][][][], [][[][]], [[][][]], [[][]][]; ... - Yann Ponty (ponty(AT)lri.fr), May 30 2007 Equals right and left borders and row sums of triangle A144218 with offset variations. - Gary W. Adamson, Sep 14 2008 The sequence is self-generated from top row A going to the left starting (1,1) and bottom row = B, the same sequence but starting (0,1) and going to the right. Take dot product of A and B and add the result to n-th term of A to get the (n+1)-th term of A. Example: a(5) = 21 as follows: Take dot product of A = (9, 4, 2, 1, 1) and (0, 1, 1, 2, 4) = (0, + 4 + 2 + 2 + 4) = 12; which is added to 9 = 21. - Gary W. Adamson, Oct 27 2008 Equals A005773 / A005773 shifted (i.e., (1,2,5,13,35,96,...) / (1,1,2,5,13,35,96,...)). - Gary W. Adamson, Dec 21 2008 Starting with offset 1 = iterates of M * [1,1,0,0,0,...], where M = a tridiagonal matrix with [0,1,1,1,...] in the main diagonal and [1,1,1,...] in the super and subdiagonals. - Gary W. Adamson, Jan 07 2009 a(n) is the number of involutions of {1,2,...,n} having genus 0. The genus g(p) of a permutation p of {1,2,...,n} is defined by g(p)=(1/2)[n+1-z(p)-z(cp')], where p' is the inverse permutation of p, c = 234...n1 = (1,2,...,n), and z(q) is the number of cycles of the permutation q. Example: a(4)=9; indeed, p=3412=(13)(24) is the only involution of {1,2,3,4} with genus > 0. This follows easily from the fact that a permutation p of {1,2,...,n} has genus 0 if and only if the cycle decomposition of p gives a noncrossing partition of {1,2,...,n} and each cycle of p is increasing (see Lemma 2.1 of the Dulucq-Simion reference). [Also, redundantly, for p=3412=(13)(24) we have cp'=2341*3412=4123=(1432) and so g(p)=(1/2)(4+1-2-1)=1.] - Emeric Deutsch, May 29 2010 Let w(i,j,n) denote walks in N^2 which satisfy the multivariate recurrence w(i,j,n) = w(i, j + 1, n - 1) + w(i - 1, j, n - 1) + w(i + 1, j - 1, n - 1) with boundary conditions w(0,0,0) = 1 and w(i,j,n) = 0 if i or j or n is < 0. Then a(n) = Sum_{i = 0..n,j = 0..n} w(i,j,n) is the number of such walks of length n. - Peter Luschny, May 21 2011 a(n)/a(n-1) tends to 3.0 as N->infinity: (1+2*cos(2*Pi/N)) relating to longest odd N regular polygon diagonals, by way of example, N=7: Using the tridiagonal generator [cf. comment of Jan 07 2009], for polygon N=7, we extract an (N-1)/2 = 3 X 3 matrix, [0,1,0; 1,1,1; 0,1,1] with an e-val of 2.24697...; the longest Heptagon diagonal with edge = 1. As N tends to infinity, the diagonal lengths tend to 3.0, the convergent of the sequence. - Gary W. Adamson, Jun 08 2011 Number of (n+1)-length permutations avoiding the pattern 132 and the dotted pattern 23\dot{1}. - Jean-Luc Baril, Mar 07 2012 Number of n-length words w over alphabet {a,b,c} such that for every prefix z of w we have #(z,a) >= #(z,b) >= #(z,c), where #(z,x) counts the letters x in word z. The a(4) = 9 words are: aaaa, aaab, aaba, abaa, aabb, abab, aabc, abac, abca. - Alois P. Heinz, May 26 2012 Number of length-n restricted growth strings (RGS) [r(1), r(2), ..., r(n)] such that r(1)=1, r(k)<=k, and r(k)!=r(k-1); for example, the 9 RGS for n=4 are 1010, 1012, 1201, 1210, 1212, 1230, 1231, 1232, 1234. - Joerg Arndt, Apr 16 2013 Number of length-n restricted growth strings (RGS) [r(1), r(2), ..., r(n)] such that r(1)=0, r(k)<=k and r(k)-r(k-1) != 1; for example, the 9 RGS for n=4 are 0000, 0002, 0003, 0004, 0022, 0024, 0033, 0222, 0224. - Joerg Arndt, Apr 17 2013 Number of (4231,5276143)-avoiding involutions in S_n. - Alexander Burstein, Mar 05 2014 a(n) is the number of increasing unary-binary trees with n nodes who have an associated permutation avoids 132. For more information about unary-binary trees with associated permutations, see A245888. - Manda Riehl, Aug 07 2014 a(n) is the number of involutions on [n] avoiding the single pattern p, where p is any one of the 8 (classical) patterns 1234, 1243, 1432, 2134, 2143, 3214, 3412, 4321. Also, number of (3412,2413)-, (3412,3142)-, (3412,2413,3142)-avoiding involutions on [n] because each of these 3 sets actually coincides with the 3412-avoiding involutions on [n]. This is a complete list of the 8 singles, 2 pairs, and 1 triple of 4-letter classical patterns whose involution avoiders are counted by the Motzkin numbers. (See Barnabei et al 2011 reference.) - David Callan, Aug 27 2014 A series created using 2*a(n)+ a(n+1) has Hankel transform of F(2n), offset 3, F being the Fibonacci bisection, A001906 (Empirical observation). - Tony Foster III, Jul 28 2016 A series created using 2*a(n) + 3*a(n+1) + a(n+2) gives the Hankel transform of Sum_{k=0..n} k*Fibonacci(2*k), offset 3, A197649 (empirical observation). - Tony Foster III, Jul 28 2016 Conjecture: (2/n)*Sum_{k=1..n} (2k+1)a(k)^2 is an integer for each positive integer n. - Zhi-Wei Sun, Nov 16 2017 The Rubey and Stump reference proves a refinement of a conjecture of René Marczinzik, which they state as: "The number of 2-Gorenstein algebras which are Nakayama algebras with n simple modules and have an oriented line as associated quiver equals the number of Motzkin paths of length n." - Eric M. Schmidt, Dec 16 2017 Number of U_{k}-equivalence classes of Łukasiewicz paths. Łukasiewicz paths are P-equivalent iff the positions of pattern P are identical in these paths. - Sergey Kirgizov, Apr 08 2018 If tau_1 and tau_2 are two distinct permutation patterns chosen from the set {132,231,312}, then a(n) is the number of valid hook configurations of permutations of [n+1] that avoid the patterns tau_1 and tau_2. - Colin Defant, Apr 28 2019 Number of permutations of length n that are sorted to the identity by a consecutive-321-avoiding stack followed by a classical-21-avoiding stack. - Colin Defant, Aug 29 2020 From Helmut Prodinger, Dec 13 2020: (Start) a(n) is the number of paths in the first quadrant starting at (0,0) and consisting of n steps from the infinite set {(1,1), (1,-1), (1,-2), (1,-3), ...}. For example, denoting U=(1,1), D=(1,-1), D_ j=(1,-j) for j >= 2, a(4) counts UUUU, UUUD, UUUD_2, UUUD_3, UUDU, UUDD, UUD_2U, UDUU, UDUD. This step set is inspired by {(1,1), (1,-1), (1,-3), (1,-5), ...}, suggested by Emeric Deutsch around 2000. See Prodinger link that contains a bijection to Motzkin paths. (End) Named by Donaghey (1977) after the Israeli-American mathematician Theodore Motzkin (1908-1970). In Sloane's "A Handbook of Integer Sequences" (1973) they were called "generalized ballot numbers". - Amiram Eldar, Apr 15 2021 Number of Motzkin n-paths a(n) is split into A107587(n), number of even Motzkin n-paths, and A343386(n), number of odd Motzkin n-paths. The value A107587(n) - A343386(n) can be called the "shadow" of a(n) (see A343773). - Gennady Eremin, May 17 2021 Conjecture: If p is a prime of the form 6m+1 (A002476), then a(p-2) is divisible by p. Currently, no counterexample exists for p < 10^7. Personal communication from Robert Gerbicz: mod such p this is equivalent to A066796 with comment: "Every A066796(n) from A066796((p-1)/2) to A066796(p-1) is divisible by prime p of form 6m+1". - Serge Batalov, Feb 08 2022 From Peter Bala, Feb 10 2022: (Start) Conjectures: (1) For prime p == 1 (mod 6) and n, r >= 1, a(n*p^r - 2) == -A005717(n-1) (mod p), where we take A005717(0) = 0 to match Batalov's conjecture above. (2) For prime p == 5 (mod 6) and n >= 1, a(n*p - 2) == -A005773(n) (mod p). (3) For prime p >= 3 and k >= 1, a(n + p^k) == a(n) (mod p) for 0 <= n <= (p^k - 3). (4) For prime p >= 5 and k >= 2, a(n + p^k) == a(n) (mod p^2) for 0 <= n <= (p^(k-1) - 3). (End) The Hankel transform of this sequence with a(0) omitted gives the period-6 sequence [1, 0, -1, -1, 0, 1, ...] which is A010892 with its first term omitted, while the Hankel transform of the current sequence is the all-ones sequence A000012, and also it is the unique sequence with this property which is similar to the unique Hankel transform property of the Catalan numbers. - Michael Somos, Apr 17 2022 REFERENCES E. Barcucci, R. Pinzani, R. Sprugnoli, The Motzkin family, P.U.M.A. Ser. A, Vol. 2, 1991, No. 3-4, pp. 249-279. P. 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. F. Bergeron, L. Favreau and D. Krob, Conjectures on the enumeration of tableaux of bounded height, Discrete Math, vol. 139, no. 1-3 (1995), 463-468. F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999) 73-112. R Bojicic, MD Petkovic, Orthogonal Polynomials Approach to the Hankel Transform of Sequences Based on Motzkin Numbers, Bulletin of the Malaysian Mathematical Sciences, 2015, doi:10.1007/s40840-015-0249-3. Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, pages 24, 298, 618, 912. Alin Bostan, Calcul Formel pour la Combinatoire des Marches, Habilitation à Diriger des Recherches, Laboratoire d’Informatique de Paris Nord, Université Paris 13, December 2017; https://specfun.inria.fr/bostan/HDR.pdf A. J. Bu, Automated counting of restricted Motzkin paths, Enumerative Combinatorics and Applications, ECA 1:2 (2021) Article S2R12. Naiomi Cameron, JE McLeod, Returns and Hills on Generalized Dyck Paths, Journal of Integer Sequences, Vol. 19, 2016, #16.6.1. L. Carlitz, Solution of certain recurrences, SIAM J. Appl. Math., 17 (1969), 251-259. Gi-Sang Cheon, S.-T. Jin, L. W. Shapiro, A combinatorial equivalence relation for formal power series, Linear Algebra and its Applications, Available online Mar 30 2015. Dairyko, Michael; Tyner, Samantha; Pudwell, Lara; Wynn, Casey. Non-contiguous pattern avoidance in binary trees. Electron. J. Combin. 19 (2012), no. 3, Paper 22, 21 pp. MR2967227. D. E. Davenport, L. W. Shapiro and L. C. Woodson, The Double Riordan Group, The Electronic Journal of Combinatorics, 18(2) (2012), #P33. E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265. T. Doslic, D. Svrtan and D. Veljan, Enumerative aspects of secondary structures, Discr. Math., 285 (2004), 67-82. Doslic, Tomislav and Veljan, Darko. Logarithmic behavior of some combinatorial sequences. Discrete Math. 308 (2008), no. 11, 2182--2212. MR2404544 (2009j:05019). S. Dulucq and R. Simion, Combinatorial statistics on alternating permutations, J. Algebraic Combinatorics, 8, 1998, 169-191. Dziemianczuk, M. "Enumerations of plane trees with multiple edges and Raney lattice paths." Discrete Mathematics 337 (2014): 9-24. Wenjie Fang, A partial order on Motzkin paths, Discrete Math., 343 (2020), #111802. I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (5.2.10). N. S. S. Gu, N. Y. Li and T. Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221. Kris Hatch, Presentation of the Motzkin Monoid, Senior Thesis, Univ. Cal. Santa Barbara, 2012; http://ccs.math.ucsb.edu/senior-thesis/Kris-Hatch.pdf. V. Jelinek, T. Mansour and M. Shattuck, On multiple pattern avoiding set partitions, Advances in Applied Mathematics Volume 50, Issue 2, February 2013, pp. 292-326. Hana Kim, R. P. Stanley A refined enumeration of hex trees and related polynomials, http://www-math.mit.edu/~rstan/papers/hextrees.pdf, Preprint 2015. S. Kitaev, Patterns in Permutations and Words, Springer-Verlag, 2011. see p. 399 Table A.7 A. Kuznetsov et al., Trees associated with the Motzkin numbers, J. Combin. Theory, A 76 (1996), 145-147. T. Lengyel, On divisibility properties of some differences of Motzkin numbers, Annales Mathematicae et Informaticae, 41 (2013) pp. 121-136. W. A. Lorenz, Y. Ponty, P. Clote, Asymptotics of RNA Shapes, Journal of Computational Biology. 2008, 15(1): 31-63. doi:10.1089/cmb.2006.0153. 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; http://www.mat.unisi.it/newsito/puma/public_html/22_2/manara_perelli-cippo.pdf. T. Mansour, Restricted 1-3-2 permutations and generalized patterns, Annals of Combin., 6 (2002), 65-76. Mansour, Toufik; Schork, Matthias; Shattuck, Mark. Catalan numbers and pattern restricted set partitions. Discrete Math. 312(2012), no. 20, 2979--2991. MR2956089. T. S. Motzkin, Relations between hypersurface cross ratios and a combinatorial formula for partitions of a polygon, for permanent preponderance and for non-associative products, Bull. Amer. Math. Soc., 54 (1948), 352-360. Jocelyn Quaintance and Harris Kwong, A combinatorial interpretation of the Catalan and Bell number difference tables, Integers, 13 (2013), #A29. J. Riordan, Enumeration of plane trees by branches and endpoints, J. Combin. Theory, A 23 (1975), 214-222. E. Royer, Interpretation combinatoire des moments negatifs des valeurs de fonctions L au bord de la bande critique, Ann. Sci. Ecole Norm. Sup. (4) 36 (2003), no. 4, 601-620. A. Sapounakis et al., Ordered trees and the inorder transversal, Disc. Math., 306 (2006), 1732-1741. A. Sapounakis, I. Tasoulas and P. Tsikouras, Counting strings in Dyck paths, Discrete Math., 307 (2007), 2909-2924. E. Schroeder, Vier combinatorische Probleme, Z. f. Math. Phys., 15 (1870), 361-376. L. W. Shapiro et al., The Riordan group, Discrete Applied Math., 34 (1991), 229-239. M. Shattuck, On the zeros of some polynomials with combinatorial coefficients, Annales Mathematicae et Informaticae, 42 (2013) pp. 93-101, http://ami.ektf.hu. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1. R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 6.37. Also Problem 7.16(b), y_3(n). P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26 (1979), 261-272. 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; http://math.nju.edu.cn/~zwsun/142p.pdf. Wang, Chenying, Piotr Miska, and István Mező. "The r-derangement numbers." Discrete Mathematics 340.7 (2017): 1681-1692. Ying Wang, Guoce Xin, A Classification of Motzkin Numbers Modulo 8, Electron. J. Combin., 25(1) (2018), #P1.54. Wen-Jin Woan, A combinatorial proof of a recursive relation of the Motzkin sequence by lattice paths. Fibonacci Quart. 40 (2002), no. 1, 3-8. Wen-jin Woan, A Recursive Relation for Weighted Motzkin Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.1.6. F. Yano and H. Yoshida, Some set partition statistics in non-crossing partitions and generating functions, Discr. Math., 307 (2007), 3147-3160. LINKS N. J. A. Sloane and Seiichi Manyama, Table of n, a(n) for n = 0..2106 (first 501 terms from N. J. A. Sloane) M. Abrate, S. Barbero, U. Cerruti, N. Murru, Fixed Sequences for a Generalization of the Binomial Interpolated Operator and for some Other Operators, J. Int. Seq. 14 (2011) # 11.8.1. M. Aigner, Motzkin Numbers, Europ. J. Comb. 19 (1998), 663-675. M. Aigner, Enumeration via ballot numbers, Discrete Math., 308 (2008), 2544-2563. J. L. Arregui, Tangent and Bernoulli numbers related to Motzkin and Catalan numbers by means of numerical triangles, arXiv:math/0109108 [math.NT], 2001. Marcello Artioli, Giuseppe Dattoli, Silvia Licciardi, Simonetta Pagnutti, Motzkin Numbers: an Operational Point of View, arXiv:1703.07262 [math.CO], 2017. Andrei Asinowski, Axel Bacher, Cyril Banderier, Bernhard Gittenberger, Analytic Combinatorics of Lattice Paths with Forbidden Patterns: Enumerative Aspects, in International Conference on Language and Automata Theory and Applications, S. Klein, C. Martín-Vide, D. Shapira (eds), Springer, Cham, pp 195-206, 2018. Andrei Asinowski, Axel Bacher, Cyril Banderier, Bernhard Gittenberger, Analytic combinatorics of lattice paths with forbidden patterns, the vectorial kernel method, and generating functions for pushdown automata, Laboratoire d'Informatique de Paris Nord (LIPN 2019). Andrei Asinowski, Cyril Banderier, Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019). A. Asinowski, G. Rote, Point sets with many non-crossing matchings, arXiv preprint arXiv:1502.04925 [cs.CG], 2015. Axel Bacher, Improving the Florentine algorithms: recovering algorithms for Motzkin and Schröder paths, arXiv:1802.06030 [cs.DS], 2018. C. Banderier, M. Bousquet-Mélou, A. Denise, P. Flajolet, D. Gardy and D. Gouyou-Beauchamps, Generating Functions for Generating Trees, Discrete Mathematics 246(1-3), March 2002, pp. 29-55. C. Banderier, C. Krattenthaler, A. Krinik, D. Kruchinin, V. Kruchinin, D. Nguyen, and M. Wallner, Explicit formulas for enumeration of lattice paths: basketball and the kernel method, arXiv preprint arXiv:1609.06473 [math.CO], 2016. Elena Barcucci, Alberto Del Lungo, Elisa Pergola, Renzo Pinzani, A methodology for plane tree enumeration, Proceedings of the 7th Conference on Formal Power Series and Algebraic Combinatorics (Noisy-le-Grand, 1995). Discrete Math. 180 (1998), no. 1-3, 45--64. MR1603693 (98m:05090). E. Barcucci et al., From Motzkin to Catalan Permutations, Discr. Math., 217 (2000), 33-49. J.-L. Baril, Classical sequences revisited with permutations avoiding dotted pattern, Electronic Journal of Combinatorics, 18 (2011), #P178. J.-L. Baril, Avoiding patterns in irreducible permutations, Discrete Mathematics and Theoretical Computer Science, Vol 17, No 3 (2016). Jean-Luc Baril, David Bevan, and Sergey Kirgizov, Bijections between directed animals, multisets and Grand-Dyck paths, arXiv:1906.11870 [math.CO], 2019. J.-L. Baril and S. Kirgizov, The pure descent statistic on permutations, 2016. J.-L. Baril and S. Kirgizov, The pure descent statistic on permutations, Discrete Mathematics, 340(10) (2017), 2550-2558. Jean-Luc Baril, Sergey Kirgizov, Armen Petrossian, Dyck paths with a first return decomposition constrained by height, 2017. Jean-Luc Baril, Sergey Kirgizov, and Armen Petrossian, Dyck paths with a first return decomposition constrained by height, Discrete Mathematics, 341(6) (2018), 1620-1628. Jean-Luc Baril, Sergey Kirgizov and Armen Petrossian, Enumeration of Łukasiewicz paths modulo some patterns, arXiv:1804.01293 [math.CO], 2018. Jean-Luc Baril, Sergey Kirgizov, Armen Petrossian, Motzkin paths with a restricted first return decomposition, Integers (2019) Vol. 19, A46. J.-L. Baril, T. Mansour, and A. Petrossian, Equivalence classes of permutations modulo excedances, 2014. J.-L. Baril and J.-M. Pallo, Motzkin subposet and Motzkin geodesics in Tamari lattices, 2013. Jean-Luc Baril, and Jean-Marcel Pallo, A Motzkin filter in the Tamari lattice, Discrete Mathematics 338(8) (2015), 1370-1378. J.-L. Baril and A. Petrossian, Equivalence classes of Dyck paths modulo some statistics, 2004. Marilena Barnabei, Flavio Bonetti, and Niccolò Castronuovo, Motzkin and Catalan Tunnel Polynomials, J. Int. Seq., Vol. 21 (2018), Article 18.8.8. Marilena Barnabei, Flavio Bonetti, and Matteo Silimbani, Restricted involutions and Motzkin paths, Advances in Applied Mathematics 47 (2011), 102-115. Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, 8 (2005), #05.4.5. 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 sequences with {-1, 0, 1} Hankel transforms, arXiv:1205.2565 [math.CO], 2012. Paul Barry, Riordan-Bernstein Polynomials, Hankel Transforms and Somos Sequences, Journal of Integer Sequences, 15 (2012), #12.8.2. Paul Barry, On the Hurwitz Transform of Sequences, Journal of Integer Sequences, 15 (2012), #12.8.7. 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), #19.5.8. Paul Barry, Characterizations of the Borel triangle and Borel polynomials, arXiv:2001.08799 [math.CO], 2020. A. M. Baxter and L. K. Pudwell, Ascent sequences avoiding pairs of patterns, preprint, 2014. A. M. Baxter and L. K. Pudwell, Ascent sequences avoiding pairs of patterns, The Electronic Journal of Combinatorics, 22(1) (2015), #P1.58. Christian Bean, Finding structure in permutation sets, Ph.D. Dissertation, Reykjavík University, School of Computer Science, 2018. Christian Bean, A Claesson, H Ulfarsson, Simultaneous Avoidance of a Vincular and a Covincular Pattern of Length 3, arXiv:1512.03226 [math.CO], 2015. Jan Bok, Graph-indexed random walks on special classes of graphs, arXiv:1801.05498 [math.CO], 2018. Miklós Bóna, Cheyne Homberger, Jay Pantone, and Vince Vatter, Pattern-avoiding involutions: exact and asymptotic enumeration, arxiv:1310.7003 [math.CO], 2013. A. Bostan, Computer Algebra for Lattice Path Combinatorics, Seminaire de Combinatoire Ph. Flajolet, Mar 28 2013. Alin Bostan, Calcul Formel pour la Combinatoire des Marches [The text is in English], Habilitation à Diriger des Recherches, Laboratoire d’Informatique de Paris Nord, Université Paris 13, December 2017. Alin Bostan and Manuel Kauers, Automatic Classification of Restricted Lattice Walks, arXiv:0811.2899 [math.CO], 2009. Alin Bostan, Andrew Elvey Price, Anthony John Guttmann, Jean-Marie Maillard, Stieltjes moment sequences for pattern-avoiding permutations, arXiv:2001.00393 [math.CO], 2020. H. Bottomley, Illustration of initial terms. 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. N. T. Cameron, Random walks, trees and extensions of Riordan group techniques, Dissertation, Howard University, 2002. Naiomi T. Cameron and Asamoah Nkwanta, On Some (Pseudo) Involutions in the Riordan Group, Journal of Integer Sequences, 8 (2005), #05.3.7. Giulio Cerbai, Anders Claesson, Luca Ferrari, Einar Steingrímsson, Sorting with pattern-avoiding stacks: the 132-machine, arXiv:2006.05692 [math.CO], 2020. J. Cigler, Some nice Hankel determinants. arXiv:1109.1449 [math.CO], 2011. Johann Cigler, Christian Krattenthaler, Hankel determinants of linear combinations of moments of orthogonal polynomials, arXiv:2003.01676 [math.CO], 2020. J. B. Cosgrave, The Gauss-Factorial Motzkin connection (Maple worksheet, change suffix to .mw). R. De Castro, A. L. Ramírez and J. L. Ramírez, Applications in Enumerative Combinatorics of Infinite Weighted Automata and Graphs, arXiv:1310.2449 [cs.DM], 2013. J. Cigler, Hankel determinants of some polynomial sequences, preprint, 2012. Colin Defant, Motzkin intervals and valid hook configurations, arXiv:1904.10451 [math.CO], 2019. Colin Defant, Troupes, Cumulants, and Stack-Sorting, arXiv:2004.11367 [math.CO], 2020. C. Defant and K. Zheng, Stack-Sorting with Consecutive-Pattern-Avoiding Stacks, arXiv:2008.12297 [math.CO], 2020. E. Deutsch and B. E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, J. Num. Theory 117 (2006), 191-215. R. M. Dickau, Delannoy and Motzkin Numbers. R. M. Dickau, The 9 paths in a 4 X 4 grid. Yun Ding and Rosena R. X. Du, Counting Humps in Motzkin paths, arXiv preprint arXiv:1109.2661 [math.CO], 2011. Filippo Disanto and Thomas Wiehe, Some instances of a sub-permutation problem on pattern avoiding permutations, arXiv:1210.6908 [math.CO], 2012. I. Dolinka, J. East, A. Evangelou, D. FitzGerald, N. Ham, Idempotent Statistics of the Motzkin and Jones Monoids, arXiv:1507.04838 [math.CO], 2015. I. Dolinka, J. East, and R. D. Gray, Motzkin monoids and partial Brauer monoids, arXiv:1512.02279 [math.GR], 2015. Robert Donaghey, Automorphisms on Catalan trees and bracketings, Journal of Combinatorial Theory, Series B, Vol. 29, No. 1 (August 1980), pp. 75-90. Robert Donaghey and Louis W. Shapiro, Motzkin numbers, J. Combin. Theory, Series A, Vol. 23, No. 3 (1977), pp. 291-301. Robert W. Donley Jr, Binomial arrays and generalized Vandermonde identities, arXiv:1905.01525 [math.CO], 2019. Ivana Đurđev, Igor Dolinka, James East, Sandwich semigroups in diagram categories, arXiv:1910.10286 [math.GR], 2019. E. S. Egge, Restricted 3412-Avoiding Involutions: Continued Fractions, Chebyshev Polynomials and Enumerations, arXiv:math/0307050 [math.CO], 2003, sec. 8. Gennady Eremin, Generating function for Naturalized Series: The case of Ordered Motzkin Words, arXiv:2002.08067 [math.CO], 2020. Gennady Eremin, Naturalized bracket row and Motzkin triangle, arXiv:2004.09866 [math.CO], 2020. Gennady Eremin, Walking in the OEIS: From Motzkin numbers to Fibonacci numbers. The "shadows" of Motzkin numbers, arXiv:2108.10676 [math.CO], 2021. Jackson Evoniuk, Steven Klee, and Van Magnan, Enumerating Minimal Length Lattice Paths, J. Int. Seq., 21 (2018), #18.3.6. 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 pages 68 and 81. Rigoberto Flórez, Leandro Junes, and José L. Ramírez, Further Results on Paths in an n-Dimensional Cubic Lattice, Journal of Integer Sequences, 21 (2018), #18.1.2. Juan B. Gil and Jordan O. Tirrell, A simple bijection for classical and enhanced k-noncrossing partitions, arXiv:1806.09065 [math.CO], 2018. Also Discrete Mathematics (2019), Article 111705. doi:10.1016/j.disc.2019.111705 Samuele Giraudo, Tree series and pattern avoidance in syntax trees, arXiv:1903.00677 [math.CO], 2019. Samuele Giraudo, The combinator M and the Mockingbird lattice, arXiv:2204.03586 [math.CO], 2022. Nils Haug, T. Prellberg, and G. Siudem, Scaling in area-weighted generalized Motzkin paths, arXiv:1605.09643 [cond-mat.stat-mech], 2016. Nickolas Hein, Jia Huang, Modular Catalan Numbers, arXiv:1508.01688 [math.CO], 2015-2016. Nickolas Hein, Jia Huang, Variations of the Catalan numbers from some nonassociative binary operations, arXiv:1807.04623 [math.CO], 2018. Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011. Cheyne Homberger, Patterns in Permutations and Involutions: A Structural and Enumerative Approach, arXiv preprint 1410.2657 [math.CO], 2014. Anders Hyllengren, Letter to N. J. A. Sloane, Oct 04 1985 Anders Hyllengren, Four integer sequences, Oct 04 1985. Observes essentially that A000984 and A002426 are inverse binomial transforms of each other, as are A000108 and A001006. INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 50 Manuel Kauers, Doron Zeilberger, Counting Standard Young Tableaux With Restricted Runs, arXiv:2006.10205 [math.CO], 2020. D. E. Knuth, Letter to L. W. Shapiro, R. K. Guy. N. J. A. Sloane, R. P. Stanley, H. Wilf regarding A001006 and A005043, Jan 18, 1989. Dmitry V. Kruchinin and Vladimir V. Kruchinin, A Generating Function for the Diagonal T_{2n,n} in Triangles, Journal of Integer Sequences, 18 (2015), #15.4.6. Marie-Louise Lackner and M. Wallner, An invitation to analytic combinatorics and lattice path counting, preprint, 2015. J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5. W. A. Lorenz, Y. Ponty and P. Clote, Asymptotics of RNA Shapes, Journal of Computational Biology 15(1) (2008), 31-63. K. Manes, A. Sapounakis, I. Tasoulas, and P. Tsikouras, Equivalence classes of ballot paths modulo strings of length 2 and 3, arXiv:1510.01952 [math.CO], 2015. Toufik Mansour, Statistics on Dyck Paths, Journal of Integer Sequences, 9 (2006), #06.1.5. T. Mansour, Restricted 1-3-2 permutations and generalized patterns, arXiv:math/0110039 [math.CO], 2001. V. Mazorchuk and B. Steinberg, Double Catalan monoids, arXiv:1105.5313 [math.GR], 2011. Peter McCalla and Asamoah Nkwanta, Catalan and Motzkin Integral Representations, arXiv:1901.07092 [math.NT], 2019. Cam McLeman and Erin McNicholas, Graph Invertibility, arXiv:1108.3588 [math.CO], 2011. Zhousheng Mei, Suijie Wang, Pattern Avoidance of Generalized Permutations, arXiv:1804.06265 [math.CO], 2018. D. Merlini, D. G. Rogers, R. Sprugnoli and M. C. Verri, On some alternative characterizations of Riordan arrays, Canad. J. Math., 49 (1997), 301-320. T. Motzkin, The hypersurface cross ratio, Bull. Amer. Math. Soc., 51 (1945), 976-984. Heinrich Niederhausen, Inverses of Motzkin and Schroeder Paths, arXiv:1105.3713 [math.CO], 2011. J.-C. Novelli and J.-Y. Thibon, Noncommutative Symmetric Functions and Lagrange Inversion, arXiv:math/0512570 [math.CO], 2005-2006. Roy Oste and Joris Van der Jeugt, Motzkin paths, Motzkin polynomials and recurrence relations, The Electronic Journal of Combinatorics, 22(2) (2015), #P2.8. Ran Pan, Dun Qiu, Jeffrey Remmel, Counting Consecutive Pattern Matches in S_n(132) and S_n(123), arXiv:1809.01384 [math.CO], 2018. Ville H. Pettersson, Enumerating Hamiltonian Cycles, The Electronic Journal of Combinatorics, 21(4) (2014), #P4.7. Simon Plouffe, The first 4431 terms. Helmut Prodinger, Deutsch paths and their enumeration, arXiv:2003.01918 [math.CO], 2020. L. Pudwell, Pattern avoidance in trees (slides from a talk, mentions many sequences), 2012. L. Pudwell, A. Baxter, Ascent sequences avoiding pairs of patterns, 2014 L. Pudwell, Pattern-avoiding ascent sequences, Slides from a talk, 2015 José L. Ramírez, The Pascal Rhombus and the Generalized Grand Motzkin Paths, arXiv:1511.04577 [math.CO], 2015. J. L. Ramírez and V. F. Sirvent, A Generalization of the k-Bonacci Sequence from Riordan Arrays, The Electronic Journal of Combinatorics, 22(1) (2015), #P1.38. Alon Regev, Amitai Regev, Doron Zeilberger, Identities in character tables of S_n, arXiv:1507.03499 [math.CO], 2015. John Riordan, Letter to N. J. A. Sloane, 1974. Dan Romik, Some formulas for the central trinomial and Motzkin numbers, J. Integer Seqs., 6 (2003). E. Rowland, R. Yassawi, Automatic congruences for diagonals of rational functions, arXiv:1310.8635 [math.NT], 2013. E. Rowland, D. Zeilberger, A Case Study in Meta-AUTOMATION: AUTOMATIC Generation of Congruence AUTOMATA For Combinatorial Sequences, arXiv:1311.4776 [math.CO], 2013. Martin Rubey and Christian Stump, Double deficiencies of Dyck paths via the Billey-Jockusch-Stanley bijection, arXiv:1708.05092 [math.CO], 2017. J. Salas and A. D. Sokal, Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. V. Further Results for the Square-Lattice Chromatic Polynomial, J. Stat. Phys. 135 (2009) 279-373, arXiv preprint, arXiv:0711.1738 [cond-mat.stat-mech], 2007-2009. Mentions this sequence. A. Sapounakis and P. Tsikouras, On k-colored Motzkin words, Journal of Integer Sequences, 7 (2004), #04.2.5. E. Schröder, Vier combinatorische Probleme, Z. f. Math. Phys., 15 (1870), 361-376. [Annotated scanned copy] Paolo Serafini, An Iterative Scheme to Compute Size Probabilities in Random Graphs and Branching Processes, Scientific Programming (2018), Article ID 3791075. N. J. A. Sloane, Illustration of initial terms. N. J. A. Sloane, Classic Sequences. N. J. A. Sloane, An Application of the OEIS (Vugraph from a talk about the OEIS). 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, 3 (2000), #00.1. Hua Sun, Yi Wang, A Combinatorial Proof of the Log-Convexity of Catalan-Like Numbers, J. Int. Seq. 17 (2014), #14.5.2 Yidong Sun and Fei Ma, Minors of a Class of Riordan Arrays Related to Weighted Partial Motzkin Paths, arXiv:1305.2015 [math.CO], 2013. Zhi-Wei Sun, Conjectures involving arithmetical sequences, in: Number Theory: Arithmetic in Shangri-La (eds., S. Kanemitsu, H. Li and J. Liu), Proc. 6th China-Japan Seminar (Shanghai, August 15-17, 2011), World Sci., Singapore, 2013, pp. 244-258. L. Takacs, Enumeration of rooted trees and forests, Math. Scientist 18 (1993), 1-10, esp. Eq. (6). Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016. P. 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. Jonas Wahl, Traces On Diagram Algebras I: Free Partition Quantum Groups, Random Lattice Paths And Random Walks On Trees, arXiv:2006.07312 [math.PR], 2020. Chen Wang, Zhi-Wei Sun, Congruences involving central trinomial coefficients, arXiv:1910.06850 [math.NT], 2019. Y. Wang, Z.-H. Zhang, Combinatorics of Generalized Motzkin Numbers, J. Int. Seq. 18 (2015), #15.2.4. Yi Wang and Bao-Xuan Zhu, Proofs of some conjectures on monotonicity of number-theoretic and combinatorial sequences, arXiv preprint arXiv:1303.5595 [math.CO], 2013. Eric Weisstein's World of Mathematics, Motzkin Number. W.-J. Woan, Hankel Matrices and Lattice Paths, J. Integer Sequences, 4 (2001), #01.1.2. J. Y. X. Yang, M. X. X. Zhong, R. D. P. Zhou, On the Enumeration of (s, s+ 1, s+2)-Core Partitions, arXiv:1406.2583 [math.CO], 2014. Huan Xiong, The number of simultaneous core partitions, arXiv:1409.7038 [math.CO], 2014. Yan X. Zhang, Four Variations on Graded Posets, arXiv:1508.00318 [math.CO], 2015. Zhuang, Yan. A generalized Goulden-Jackson cluster method and lattice path enumeration, Discrete Mathematics 341.2 (2018): 358-379. Also arXiv: 1508.02793v2. FORMULA G.f.: A(x) = ( 1 - x - (1-2*x-3*x^2)^(1/2) ) / (2*x^2). G.f. A(x) satisfies A(x) = 1 + x*A(x) + x^2*A(x)^2. G.f.: F(x)/x where F(x) is the reversion of x/(1+x+x^2). - Joerg Arndt, Oct 23 2012 a(n) = (-1/2) Sum_{i+j = n+2, i >= 0, j >= 0} (-3)^i*C(1/2, i)*C(1/2, j). a(n) = (3/2)^(n+2) * Sum_{k >= 1} 3^(-k) * Catalan(k-1) * binomial(k, n+2-k). [Doslic et al.] a(n) ~ 3^(n+1)*sqrt(3)*(1 + 1/(16*n))/((2*n+3)*sqrt((n+2)*Pi)). [Barcucci, Pinzani and Sprugnoli] Limit_{n->infinity} a(n)/a(n-1) = 3. [Aigner] a(n+2) - a(n+1) = a(0)*a(n) + a(1)*a(n-1) + ... + a(n)*a(0). [Bernhart] a(n) = (1/(n+1)) * Sum_{i} (n+1)!/(i!*(i+1)!*(n-2*i)!). [Bernhart] From Len Smiley: (Start) a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*A000108(k+1). a(n) = (1/(n+1))*Sum_{k=0..ceiling((n+1)/2)} binomial(n+1, k)*binomial(n+1-k, k-1); D-finite with recurrence: (n+2)*a(n) = (2*n+1)*a(n-1) + (3*n-3)*a(n-2). (End) a(n) = Sum_{k=0..n} C(n, 2k)*A000108(k). - Paul Barry, Jul 18 2003 E.g.f.: exp(x)*BesselI(1, 2*x)/x. - Vladeta Jovovic, Aug 20 2003 a(n) = A005043(n) + A005043(n+1). The Hankel transform of this sequence gives A000012 = [1, 1, 1, 1, 1, 1, ...]. E.g., Det([1, 1, 2, 4; 1, 2, 4, 9; 2, 4, 9, 21; 4, 9, 21, 51]) = 1. - Philippe Deléham, Feb 23 2004 a(m+n) = Sum_{k>=0} A064189(m, k)*A064189(n, k). - Philippe Deléham, Mar 05 2004 a(n) = (1/(n+1))*Sum_{j=0..floor(n/3)} (-1)^j*binomial(n+1, j)*binomial(2*n-3*j, n). - Emeric Deutsch, Mar 13 2004 a(n) = A086615(n) - A086615(n-1) (n >= 1). - Emeric Deutsch, Jul 12 2004 G.f.: A(x)=(1-y+y^2)/(1-y)^2 where (1+x)*(y^2-y)+x=0; A(x)=4*(1+x)/(1+x+sqrt(1-2*x-3*x^2))^2; a(n)=(3/4)*(1/2)^n*Sum_(k=0..2*n, 3^(n-k)*C(k)*C(k+1, n+1-k) ) + 0^n/4 [after Doslic et al.]. - Paul Barry, Feb 22 2005 G.f.: c(x^2/(1-x)^2)/(1-x), c(x) the g.f. of A000108. - Paul Barry, May 31 2006 Asymptotic formula: a(n) ~ sqrt(3/4/Pi)*3^(n+1)/n^(3/2). - Benoit Cloitre, Jan 25 2007 a(n) = A007971(n+2)/2. - Zerinvary Lajos, Feb 28 2007 a(n) = (1/(2*Pi))*Integral_{x=-1..3} x^n*sqrt((3-x)*(1+x)) is the moment representation. - Paul Barry, Sep 10 2007 Equals inverse binomial transform of A000108 starting (1, 2, 5, 14, 42, ...). - Gary W. Adamson, Dec 10 2007 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() is the Catalan numbers A000108. The present sequence is Phi([0,1,1]), see the 6th formula. - Gary W. Adamson, Oct 27 2008 G.f.: 1/(1-x-x^2/(1-x-x^2/(1-x-x^2/(1-x-x^2/(1-x-x^2/.... (continued fraction). - Paul Barry, Dec 06 2008 G.f.: 1/(1-(x+x^2)/(1-x^2/(1-(x+x^2)/(1-x^2/(1-(x+x^2)/(1-x^2/(1-.... (continued fraction). - Paul Barry, Feb 08 2009 a(n) = (-3)^(1/2)/(6*(n+2)) * (-1)^n*(3*hypergeom([1/2, n+1],,4/3) - hypergeom([1/2, n+2],,4/3)). - Mark van Hoeij, Nov 12 2009 G.f.: 1/(1-x/(1-x/(1-x^2/(1-x/(1-x/(1-x^2/(1-x/(1-x/(1-x^2/(1-... (continued fraction). - Paul Barry, Mar 02 2010 G.f.: 1/(1-x/(1-x/(1+x-x/(1-x/(1+x-x/(1-x/(1+x-x/(1-x/(1+x-x/(1-... (continued fraction). - Paul Barry, Jan 26 2011 [Adds apparently a third '1' in front. - R. J. Mathar, Jan 29 2011] Let A(x) be the g.f., then B(x)=1+x*A(x) = 1 + 1*x + 1*x^2 + 2*x^3 + 4*x^4 + 9*x^5 + ... = 1/(1-z/(1-z/(1-z/(...)))) where z=x/(1+x) (continued fraction); more generally B(x)=C(x/(1+x)) where C(x) is the g.f. for the Catalan numbers (A000108). - Joerg Arndt, Mar 18 2011 a(n) = (2/Pi)*Integral_{x=-1..1} (1+2*x)^n*sqrt(1-x^2). - Peter Luschny, Sep 11 2011 G.f.: (1-x-sqrt(1-2*x-3*(x^2)))/(2*(x^2)) = 1/2/(x^2)-1/2/x-1/2/(x^2)*G(0); G(k) = 1+(4*k-1)*x*(2+3*x)/(4*k+2-x*(2+3*x)*(4*k+1)*(4*k+2) /(x*(2+3*x)*(4*k+1)+(4*k+4)/G(k+1)), if -1 < x < 1/3; (continued fraction). - Sergei N. Gladkovskii, Dec 01 2011 G.f.: (1-x-sqrt(1-2*x-3*(x^2)))/(2*(x^2)) = (-1 + 1/G(0))/(2*x); G(k) = 1-2*x/(1+x/(1+x/(1-2*x/(1-x/(2-x/G(k+1)))))); (continued fraction). - Sergei N. Gladkovskii, Dec 11 2011 0 = a(n) * (9*a(n+1) + 15*a(n+2) - 12*a(n+3)) + a(n+1) * ( -3*a(n+1) + 10*a(n+2) - 5*a(n+3)) + a(n+2) * (a(n+2) + a(n+3)) unless n=-2. - Michael Somos, Mar 23 2012 a(n) = (-1)^n*hypergeometric([-n,3/2],,4). - Peter Luschny, Aug 15 2012 Representation in terms of special values of Jacobi polynomials P(n,alpha,beta,x), in Maple notation: a(n)= 2*(-1)^n*n!*JacobiP(n,2,-3/2-n,-7)/(n+2)!, n>=0. - Karol A. Penson, Jun 24 2013 G.f.: Q(0)/x - 1/x, where Q(k) = 1 + (4*k+1)*x/((1+x)*(k+1) - x*(1+x)*(2*k+2)*(4*k+3)/(x*(8*k+6)+(2*k+3)*(1+x)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 14 2013 Catalan(n+1) = Sum_{k=0..n} binomial(n,k)*a(k). E.g.: 42 = 1*1 + 4*1 + 6*2 + 4*4 + 1*9. - Doron Zeilberger, Mar 12, 2015 G.f. A(x) with offset 1 satisfies: A(x)^2 = A( x^2/(1-2*x) ). - Paul D. Hanna, Nov 08 2015 Conjecture: +(n+2)*a(n) +(-2*n-1)*a(n-1) +3*(-n+1)*a(n-2)=0. - R. J. Mathar, Sep 06 2016 [Conjecture follows from the D.E. (3*x^3+2*x^2-x)*g'(x)+(3*x^2+3*x-2)*g(x)+2=0 satisfied by the g.f. - Robert Israel, Mar 16 2018] a(n) = GegenbauerPoly(n,-n-1,-1/2)/(n+1). - Emanuele Munarini, Oct 20 2016 a(n) = a(n-1) + A002026(n-1). Number of Motzkin paths that start with an F step plus number of Motzkin paths that start with an U step. - R. J. Mathar, Jul 25 2017 G.f. A(x) satisfies A(x)*A(-x) = F(x^2), where F(x) is the g.f. of A168592. - Alexander Burstein, Oct 04 2017 G.f.: A(x) = exp(int((E(x)-1)/x dx), where E(x) is the g.f. of A002426. Equivalently, E(x) = 1 + x*A'(x)/A(x). - Alexander Burstein, Oct 05 2017 G.f. A(x) satisfies: A(x) = Sum_{j>=0} x^j * Sum_{k=0..j} binomial(j,k)*x^k*A(x)^k. - Ilya Gutkovskiy, Apr 11 2019 From Gennady Eremin, May 08 2021: (Start) G.f.: 2/(1 - x + sqrt(1-2*x-3*x^2)). a(n) = A107587(n) + A343386(n) = 2*A107587(n) - A343773(n) = 2*A343386(n) + A343773(n). (End) Revert transform of A049347 (after Michael Somos). - Gennady Eremin, Jun 11 2021 Sum_{n>=0} 1/a(n) = 2.941237337631025604300320152921013604885956025483079699366681494505960039781389... - Vaclav Kotesovec, Jun 17 2021 Let a(-1) = (1 - sqrt(-3))/2 and a(n) = a(-3-n)*(-3)^(n+3/2) for all n in Z. Then a(n) satisfies my previous formula relation from Mar 23 2012 now for all n in Z. - Michael Somos, Apr 17 2022 EXAMPLE G.f.: 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 21*x^5 + 51*x^6 + 127*x^7 + 323*x^8 + ... MAPLE # Three different Maple scripts for this sequence: [seq(add(binomial(n+1, k)*binomial(n+1-k, k-1), k=0..ceil((n+1)/2))/(n+1), n=0..50)]; A001006 := proc(n) option remember; local k; if n <= 1 then 1 else procname(n-1) + add(procname(k)*procname(n-k-2), k=0..n-2); fi; end; Order := 20: solve(series(x/(1+x+x^2), x)=y, x); zl:=4*(1-z+sqrt(1-2*z-3*z^2))/(1-z+sqrt(1-2*z-3*z^2))^2/2: gser:=series(zl, z=0, 35): seq(coeff(gser, z, n), n=0..29); # Zerinvary Lajos, Feb 28 2007 # n -> [a(0), a(1), .., a(n)] A001006_list := proc(n) local w, m, j, i; w := proc(i, j, n) option remember; if min(i, j, n) < 0 or max(i, j) > n then 0 elif n = 0 then if i = 0 and j = 0 then 1 else 0 fi else w(i, j + 1, n - 1) + w(i - 1, j, n - 1) + w(i + 1, j - 1, n - 1) fi end: [seq( add( add( w(i, j, m), i = 0..m), j = 0..m), m = 0..n)] end: A001006_list(29); # Peter Luschny, May 21 2011 MATHEMATICA a = 1; a[n_Integer] := a[n] = a[n - 1] + Sum[a[k] * a[n - 2 - k], {k, 0, n - 2}]; Array[a, 30] (* Second program: *) CoefficientList[Series[(1 - x - (1 - 2x - 3x^2)^(1/2))/(2x^2), {x, 0, 29}], x] (* Jean-François Alcover, Nov 29 2011 *) Table[Hypergeometric2F1[(1-n)/2, -n/2, 2, 4], {n, 0, 29}] (* Peter Luschny, May 15 2016 *) Table[GegenbauerC[n, -n-1, -1/2]/(n+1), {n, 0, 100}] (* Emanuele Munarini, Oct 20 2016 *) MotzkinNumber = DifferenceRoot[Function[{y, n}, {(-3n-3)*y[n] + (-2n-5)*y[n+1] + (n+4)*y[n+2] == 0, y == 1, y == 1}]]; Table[MotzkinNumber[n], {n, 0, 29}] (* Jean-François Alcover, Oct 27 2021 *) PROG (PARI) {a(n) = polcoeff( ( 1 - x - sqrt((1 - x)^2 - 4 * x^2 + x^3 * O(x^n))) / (2 * x^2), n)}; /* Michael Somos, Sep 25 2003 */ (PARI) {a(n) = if( n<0, 0, n++; polcoeff( serreverse( x / (1 + x + x^2) + x * O(x^n)), n))}; /* Michael Somos, Sep 25 2003 */ (PARI) {a(n) = if( n<0, 0, n! * polcoeff( exp(x + x * O(x^n)) * besseli(1, 2 * x + x * O(x^n)), n))}; /* Michael Somos, Sep 25 2003 */ (Maxima) a:1$a:1$ a[n]:=((2*n+1)*a[n-1]+(3*n-3)*a[n-2])/(n+2)$makelist(a[n], n, 0, 12); /* Emanuele Munarini, Mar 02 2011 */ (Maxima) M(n) := coeff(expand((1+x+x^2)^(n+1)), x^n)/(n+1); makelist(M(n), n, 0, 60); /* Emanuele Munarini, Apr 04 2012 */ (Maxima) makelist(ultraspherical(n, -n-1, -1/2)/(n+1), n, 0, 12); /* Emanuele Munarini, Oct 20 2016 */ (Haskell) a001006 n = a001006_list !! n a001006_list = zipWith (+) a005043_list$ tail a005043_list -- Reinhard Zumkeller, Jan 31 2012 (Python) from gmpy2 import divexact A001006 = [1, 1] for n in range(2, 10**3):     A001006.append(divexact(A001006[-1]*(2*n+1)+(3*n-3)*A001006[-2], n+2)) # Chai Wah Wu, Sep 01 2014 (Sage and Python) def mot():     a, b, n = 0, 1, 1     while True:         yield b//n         n += 1         a, b = b, (3*(n-1)*n*a+(2*n-1)*n*b)//((n+1)*(n-1)) A001006 = mot() print([next(A001006) for n in range(30)]) # Peter Luschny, May 16 2016 CROSSREFS Cf. A026300, A005717, A020474, A001850, A004148. First column of A064191, A064189, A000108, A088615, A007971, A001405, A005817, A049401, A007579, A007578, A097862, A144218, A005773, A178515, A217275. First row of A064645. Bisections: A026945, A099250. Sequences related to chords in a circle: A001006, A054726, A006533, A006561, A006600, A007569, A007678. See also entries for chord diagrams in Index file. a(n) = A005043(n)+A005043(n+1). A086246 is another version, although this is the main entry. Column k=3 of A182172. Motzkin numbers A001006 read mod 2,3,4,5,6,7,8,11: A039963, A039964, A299919, A258712, A299920, A258711, A299918, A258710. Cf. A004148, A004149, A023421, A023422, A023423, A290277 (inv. Euler Transf.). Sequence in context: A166587 A292440 A168049 * A086246 A247100 A230556 Adjacent sequences:  A001003 A001004 A001005 * A001007 A001008 A001009 KEYWORD nonn,core,easy,nice,changed AUTHOR STATUS approved

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Last modified May 23 23:07 EDT 2022. Contains 353993 sequences. (Running on oeis4.)