A004148 -id:A004148 - 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

A005251 a(0) = 0, a(1) = a(2) = a(3) = 1; thereafter, a(n) = a(n-1) + a(n-2) + a(n-4).

Original entry on oeis.org

0, 1, 1, 1, 2, 4, 7, 12, 21, 37, 65, 114, 200, 351, 616, 1081, 1897, 3329, 5842, 10252, 17991, 31572, 55405, 97229, 170625, 299426, 525456, 922111, 1618192, 2839729, 4983377, 8745217, 15346786, 26931732, 47261895, 82938844, 145547525, 255418101, 448227521

Offset: 0

N. J. A. Sloane and R. K. Guy

a(n+3) is the number of n-bit sequences that avoid 010. Example: For n=4 the 12 sequences are all 4-bit sequences except 0100, 0101, 0010, 1010. - David Callan, Mar 25 2004
a(n+2) is the number of compositions (ordered partitions) of n where no two adjacent parts are != 1, see example. - Joerg Arndt, Jan 26 2013
a(n+1) is the number of compositions of n avoiding the part 2. - Joerg Arndt, Jul 13 2014
Number of different positive braids with n crossings of 3 strands.
This is a_2(n) in the Doroslovacki reference. Note that there is a typo in the paper in the formula for a_2(n): the upper bound in the inner sum should be "n-i" not "i-1". - Max Alekseyev, Jun 26 2007
a(n) is the number of peakless Motzkin paths of length n-1 with no UHH...HD's starting at level > 0 (here n > 0 and U=(1,1), H=(1,0), D=(1,-1)). Example: a(5)=7 because from all 8 peakless Motzkin paths of length 5 (see A004148) only UUHDD does not qualify. - Emeric Deutsch, Sep 13 2004
Equals the INVERT transform of (1, 0, 1, 1, 1, ...); equivalent to a(n) = a(n-1) + a(n-3) + a(n-4) + ... - Gary W. Adamson, Apr 27 2009
a(n) is the number of length n-1 words on {0,1} such that each string of 1's is followed by a string of at least two 0's. For example, a(5) = 4 because we have: 0000, 0100, 1000, and 1100. - Geoffrey Critzer, Aug 09 2013
a(n+1) is the top left entry of the n-th power of any of the 3 X 3 matrices [1, 1, 0; 0, 1, 1; 1, 0, 0] or [1, 0, 1; 1, 1, 0; 0, 1, 0] or [1, 1, 0; 0, 0, 1; 1, 0, 1] or [1, 0, 1; 1, 0, 0; 0, 1, 1]. - R. J. Mathar, Feb 03 2014
For n >= 2, a(n) is the number of (n-2)-length binary words with no isolated zeros. - Milan Janjic, Mar 07 2015
Apart from the first three terms, the total number of bargraphs of semiperimeter n of height at most two for n >= 2 starts 1, 2, 4, 7, 12, ... - Arnold Knopfmacher, Nov 02 2016
Number of DD-equivalence classes of Łukasiewicz paths. Łukasiewicz paths are DD-equivalent iff the positions of pattern DD are identical in these paths. - Sergey Kirgizov, Apr 08 2018
From Gus Wiseman, Nov 25 2019: (Start)
For n > 0, also the number of subsets of {1, ..., n - 3} such that if x and x + 2 are both in the subset, then so is x + 1. For example, the a(3) = 1 through a(7) = 12 subsets are:
  {}  {}   {}     {}       {}
      {1}  {1}    {1}      {1}
           {2}    {2}      {2}
           {1,2}  {3}      {3}
                  {1,2}    {4}
                  {2,3}    {1,2}
                  {1,2,3}  {1,4}
                           {2,3}
                           {3,4}
                           {1,2,3}
                           {2,3,4}
                           {1,2,3,4}
(End)
The two-dimensional version, which counts sets of pairs where, if two pairs are separated by graph-distance 2, then the intermediate pair or pairs are also in the set, is A329871. - Gus Wiseman, Nov 30 2019
a(n+1) is the number of ways to tile a strip of length n with squares, dominoes, and tetrominoes, where the first tile cannot be a domino. - Greg Dresden and Myanna Nash, Aug 18 2020
For n>=3, a(n) is the number of binary strings of length n-2 without any maximal runs of ones of length 1. - Félix Balado, Aug 25 2025

			From _Joerg Arndt_, Jan 26 2013: (Start)
The a(5+2) = 12 compositions of 5 where no two adjacent parts are != 1 are
  [ 1]  [ 1 1 1 1 1 ]
  [ 2]  [ 1 1 1 2 ]
  [ 3]  [ 1 1 2 1 ]
  [ 4]  [ 1 1 3 ]
  [ 5]  [ 1 2 1 1 ]
  [ 6]  [ 1 3 1 ]
  [ 7]  [ 1 4 ]
  [ 8]  [ 2 1 1 1 ]
  [ 9]  [ 2 1 2 ]
  [10]  [ 3 1 1 ]
  [11]  [ 4 1 ]
  [12]  [ 5 ]
(End)
G.f. = x + x^2 + x^3 + 2*x^4 + 4*x^5 + 7*x^6 + 12*x^7 + 21*x^8 + 37*x^9 + ...

S. Burckel, Efficient methods for three strand braids (submitted). [Apparently unpublished]
P. Chinn and S. Heubach, "Compositions of n with no occurrence of k", Congressus Numeratium, 2002, v. 162, pp. 33-51.
John H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, p. 205.
R. K. Guy, "Anyone for Twopins?" in D. A. Klarner, editor, The Mathematical Gardner. Prindle, Weber and Schmidt, Boston, 1981, pp. 2-15.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..500
Kassie Archer, Ethan Borsh, Jensen Bridges, Christina Graves, and Millie Jeske, Cyclic permutations avoiding patterns in both one-line and cycle forms, arXiv:2312.05145 [math.CO], 2023. See p. 2.
Andrei Asinowski and Cyril Banderier, On Lattice Paths with Marked Patterns: Generating Functions and Multivariate Gaussian Distribution, 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020) Leibniz International Proceedings in Informatics (LIPIcs) Vol. 159, 1:1-1:16.
R. Austin and R. K. Guy, Binary sequences without isolated ones, Fib. Quart., 16 (1978), 84-86.
J.-L. Baril, Avoiding patterns in irreducible permutations, Discrete Mathematics and Theoretical Computer Science, Vol 17, No 3 (2016). See Table 4.
Jean-Luc Baril, Sergey Kirgizov and Armen Petrossian, Enumeration of Łukasiewicz paths modulo some patterns, arXiv:1804.01293 [math.CO], 2018.
N. Bergeron, S. Mykytiuk, F. Sottile and S. van Willigenburg, Shifted quasisymmetric functions and the Hopf algebra of peak functions, arXiv:math/9904105 [math.CO], 1999.
D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, Example 11.
A. Blecher, C. Brennan, A. Knopfmacher and H. Prodinger, The height and width of bargraphs, Discrete Applied Math. 180, (2015), 36-44.
A. Brousseau, Fibonacci and Related Number Theoretic Tables, Fibonacci Association, San Jose, CA, 1972, p. 112.
P. Chinn and S. Heubach, Integer Sequences Related to Compositions without 2's, J. Integer Seqs., Vol. 6, 2003.
James Currie, Pascal Ochem, Narad Rampersad, and Jeffrey Shallit, Properties of a Ternary Infinite Word, arXiv:2206.01776 [cs.DM], 2022.
James Currie, Pascal Ochem, Narad Rampersad, and Jeffrey Shallit, Complement Avoidance in Binary Words, arXiv:2209.09598 [math.CO], 2022.
J. Demetrovics et al., On the number of unions in a family of sets, in Combinatorial Math., Proc. 3rd Internat. Conf., Annals NY Acad. Sci., 555 (1989), 150-158.
R. Doroslovacki, Binary sequences without 011...110 (k-1 1's) for fixed k, Mat. Vesnik 46 (1994), no. 3-4, 93-98.
Nazim Fatès, Biswanath Sethi, and Sukanta Das, On the Reversibility of ECAs with Fully Asynchronous Updating: The Recurrence Point of View, in Reversibility and Universality, Andrew Adamatzky, editor, Emergence, Complexity and Computation Vol. 30. Springer, 2018.
Steven Finch, Cantor-solus and Cantor-multus distributions, arXiv:2003.09458 [math.CO], 2020.
R. L. Graham and N. J. A. Sloane, Anti-Hadamard matrices, Linear Alg. Applic., 62 (1984), 113-137.
R. K. Guy, Letter to N. J. A. Sloane, Feb 1986
R. K. Guy, Anyone for Twopins?, in D. A. Klarner, editor, The Mathematical Gardner. Prindle, Weber and Schmidt, Boston, 1981, pp. 2-15. [Annotated scanned copy, with permission]
V. C. Harris and C. C. Styles, A generalization of Fibonacci numbers, Fib. Quart. 2 (1964) 277-289, sequence u(n,1,2).
V. E. Hoggatt, Jr., 7-page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 98
Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3.
Vedran Krcadinac, A new generalization of the golden ratio, Fibonacci Quart. 44 (2006), no. 4, 335-340.
Erkko Lehtonen and Tamás Waldhauser, Associative spectra of graph algebras II. Satisfaction of bracketing identities, spectrum dichotomy, arXiv:2011.08522 [math.CO], 2020.
J. J. Madden, A generating function for the distribution of runs in binary words, arXiv:1707.04351 [math.CO], 2017, Theorem 1.1, r=2, k=0.
T. Mansour and M. Shattuck, Counting Peaks and Valleys in a Partition of a Set, J. Int. Seq. 13 (2010), 10.6.8, Lemma 2.1, k=2, 0 peaks.
Denis Neiter and Amsha Proag, Links Between Sums Over Paths in Bernoulli's Triangles and the Fibonacci Numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.8.3.
Nicolas Ollinger and Jeffrey Shallit, The Repetition Threshold for Rote Sequences, arXiv:2406.17867 [math.CO], 2024.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
A. G. Shannon, Some recurrence relations for binary sequence matrices, NNTDM 17 (2011), 4, 913.
Bojan Vučković and Miodrag Živković, Row Space Cardinalities Above 2^(n - 2) + 2^(n - 3), ResearchGate, January 2017, p. 3.
Index entries for linear recurrences with constant coefficients, signature (2,-1,1).

Cf. A001608, A004148, A005314, A006498, A011973, A049864, A049853, A078065, A118891, A173022, A176971, A178470, A261041, A303696, A329871, A384153.
Bisection of Padovan sequence A000931.
Partial sums of A005314 shifted 3 times to the right, if we assume A005314(0) = 1.
Compositions without adjacent equal parts are A003242.
Compositions without isolated parts are A114901.
Row sums of A097230(n-2) for n>1.

a005251 n = a005251_list !! n
a005251_list = 0 : 1 : 1 : 1 : zipWith (+) a005251_list
   (drop 2 $ zipWith (+) a005251_list (tail a005251_list))
-- Reinhard Zumkeller, Dec 28 2011

I:=[0,1,1,1]; [n le 4 select I[n] else Self(n-1)+Self(n-2)+Self(n-4): n in [1..45]]; // Vincenzo Librandi, Nov 30 2018

R:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x*(1-x)/(1-2*x + x^2 - x^3) )); // Marius A. Burtea, Oct 24 2019

A005251 := proc(n) option remember; if n <= 2 then n elif n = 3 then 4 else 2*A005251(n - 1) - A005251(n - 2) + A005251(n - 3); fi; end;
A005251:=(-1+z)/(-1+2*z-z**2+z**3); # Simon Plouffe in his 1992 dissertation
a := n -> `if`(n<=1, n, hypergeom([(2-n)/3, 1-n/3, (1-n)/3], [1/2, -n+1], 27/4)):
seq(simplify(a(n)), n=0..36); # Peter Luschny, Apr 08 2018

LinearRecurrence[{2,-1,1},{0,1,1},40]  (* Harvey P. Dale, May 05 2011 *)
a[ n_]:= If[n<0, SeriesCoefficient[ -x(1-x)/(1 -x + 2x^2 -x^3), {x, 0, -n}], SeriesCoefficient[ x(1-x)/(1 -2x +x^2 -x^3), {x, 0, n}]] (* Michael Somos, Dec 13 2013 *)
a[0] = 1; a[1] = a[2] = 0; a[n_] := a[n] = a[n-2] + a[n-3]; Table[a[2 n-1], {n, 1, 20}] (* Rigoberto Florez, Oct 15 2019 *)
Table[If[n==0,0,Length[DeleteCases[Subsets[Range[n-3]],{_,x_,y_,_}/;x+2==y]]],{n,0,10}] (* Gus Wiseman, Nov 25 2019 *)

Vec((1-x)/(1-2*x+x^2-x^3)+O(x^99)) /* Charles R Greathouse IV, Nov 20 2012 */

{a(n) = if( n<0, polcoeff( -x*(1-x)/(1 -x +2*x^2 -x^3) + x*O(x^-n), -n), polcoeff( x*(1-x)/(1 -2*x +x^2 -x^3) + x*O(x^n), n))} /* Michael Somos, Dec 13 2013 */

[sum( binomial(n-j-1, 2*j) for j in (0..floor((n-1)/3)) ) for n in (0..50)] # G. C. Greubel, Apr 13 2022

a(n) = 2*a(n-1) - a(n-2) + a(n-3).
G.f.: z*(1-z)/(1 - 2*z + z^2 - z^3). - Emeric Deutsch, Sep 13 2004
23*a_n = 3*P_{2n+1} + 7*P_{2n} - 2*P_{2n-1}, where P_n are the Perrin numbers, A001608. - Don Knuth, Dec 09 2008
a(n+1) = Sum_{k=0..n} binomial(n-k, 2k). - Richard L. Ollerton, May 12 2004
From Henry Bottomley, Feb 21 2001: (Start)
a(n) = (Sum_{ja(n) = A005314(n) - A005314(n-1).
a(n) = A049853(n-1) - a(n-1).
a(n) = A005314(n) - a(n-2). (End)
Conjecture: a(n+1) + |A078065(n)| = 2*A005314(n+1). - Creighton Dement, Dec 21 2004
a(n+2) has g.f. (F_3(-x) + F_2(-x))/(F_4(-x) + F_3(-x)) = 1/(-x+1/(-x+1/(-x+1))) where F_n(x) is the n-th Fibonacci polynomial; see A011973. - Qiaochu Yuan (qchu(AT)mit.edu), Feb 19 2009
a(n) = A173022(2^(n-2) - 1) for n > 1. - Reinhard Zumkeller, Feb 07 2010
BINOMIAL transform of A176971 is a(n+1). - Michael Somos, Dec 13 2013
a(n) = hypergeom([(2-n)/3, 1-n/3, (1-n)/3], [1/2, -n+1], 27/4) for n > 1. - Peter Luschny, Apr 08 2018
G.f.: z/(1-z-z^3-z^4-z^5-...) for the compositions of n-1 avoiding 2. The g.f. for the number of compositions of n avoiding the part k is 1/(1-z-...-z^(k-1) - z^(k+1)-...). - Gregory L. Simay, Sep 09 2018
If p,q,r are the three solutions to x^3 = 2x^2 - x + 1, then a(n) = (p-1)*p^n/((p-q)*(p-r)) + (q-1)*q^n/((q-p)*(q-r)) + (r-1)*r^n/((r-p)*(r-q)). - Greg Dresden and AnXing Yang, Aug 12 2025

A110320 Number of blocks in all RNA secondary structures with n nodes (an RNA secondary structure can be viewed as a restricted noncrossing partition).

Original entry on oeis.org

1, 2, 5, 13, 32, 80, 201, 505, 1273, 3217, 8146, 20668, 52531, 133726, 340909, 870213, 2223958, 5689807, 14571335, 37350585, 95821071, 246015677, 632088930, 1625119218, 4180845277, 10762096850, 27718352411, 71426753423, 184146711578

Offset: 1

Emeric Deutsch, Jul 19 2005

nonn

Antidiagonal sums of A132812. - Philippe Deléham, Jun 08 2013

			a(4)=13 because the 4 (=A004148(4)) RNA secondary structures of size 4, namely 1/2/3/4, 13/2/4, 14/2/3 and 1/24/3, have altogether 4+3+3+3=13 blocks.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Jean-Luc Baril, Sergey Kirgizov, Rémi Maréchal, and Vincent Vajnovszki, Grand Dyck paths with air pockets, arXiv:2211.04914 [math.CO], 2022.
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
Peter McCalla and Asamoah Nkwanta, Catalan and Motzkin Integral Representations, arXiv:1901.07092 [math.NT], 2019.
W. R. Schmitt and M. S. Waterman, Linear trees and RNA secondary structure, Discrete Appl. Math., 51, 317-323, 1994.
P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26 (1978), 261-272.
M. Vauchassade de Chaumont and G. Viennot, Polynomes orthogonaux et problèmes d'énumeration en biologie moléculaire, Publ. I.R.M.A. Strasbourg, 1984, 229/S-08, Actes 8e Sem. Lotharingien, pp. 79-86.

Cf. A004148, A110319.

G:=1/2*(1-z-z^2)/z^2/(1-2*z-z^2-2*z^3+z^4)^(1/2)-1/2*1/(z^2): Gser:=series(G,z=0,37): seq(coeff(Gser,z^n),n=1..33);

Table[Sum[Binomial[n-j+1,j]Binomial[n-j+1,j-1],{j, 0, n}],{n,1,25}] (* Benedict W. J. Irwin, Sep 24 2016 *)

G.f.: (1-z-z^2)/(2*z^2*sqrt(1-2*z-z^2-2*z^3+z^4))-1/(2*z^2).
a(n) = Sum_{k=1..n} k*A110319(n,k).
Conjecture: a(n) = (A051292(n+2)-A051286(n+1))/2. - Gerald McGarvey, Jan 14 2007
a(n) = (A051286(n+2)-A051286(n+1)-A051286(n))/2. - Benedict W. J. Irwin, Sep 24 2016
a(n) ~ sqrt(4 + 9/sqrt(5)) * (3+sqrt(5))^n / (sqrt(Pi*n) * 2^(n+1)). - Vaclav Kotesovec, Sep 25 2016, equivalently, a(n) ~ phi^(2*n + 3) / (2 * 5^(1/4) * sqrt(Pi*n)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 06 2021
D-finite with recurrence (n+2)*a(n) +3*(-n-1)*a(n-1) +(n-7)*a(n-3) +2*(2*n-3)*a(n-4) +(n-5)*a(n-5) +(-n+4)*a(n-6)=0. - R. J. Mathar, Feb 21 2020

A023431 Generalized Catalan Numbers x^3A(x)^2 + (x-1)A(x) + 1 =0.

Original entry on oeis.org

1, 1, 1, 2, 4, 7, 13, 26, 52, 104, 212, 438, 910, 1903, 4009, 8494, 18080, 38656, 82988, 178802, 386490, 837928, 1821664, 3970282, 8673258, 18987930, 41652382, 91539466, 201525238, 444379907, 981384125, 2170416738, 4806513660

Offset: 0

Olivier Gérard

Essentially the same as A025246.
Number of lattice paths in the first quadrant from (0,0) to (n,0) using only steps H=(1,0), U=(1,1) and D=(2,-1). E.g. a(5)=7 because we have HHHHH, HHUD, HUDH, HUHD, UDHH, UHDH and UHHD. - Emeric Deutsch, Dec 25 2003
Also number of peakless Motzkin paths of length n with no double rises; in other words, Motzkin paths of length n with no UD's and no UU's, where U=(1,1) and D=(1,-1). E.g. a(5)=7 because we have HHHHH, HHUHD, HUHDH, HUHHD, UHDHH, UHHDH and UHHHD, where H=(1,0). - Emeric Deutsch, Jan 09 2004
Series reversion of g.f. A(x) is -A(-x) (if offset 1). - Michael Somos, Jul 13 2003
Hankel transform is A010892(n+1). [From Paul Barry, Sep 19 2008]
Number of FU_{k}-equivalence classes of Łukasiewicz paths. Łukasiewicz paths are P-equivalent iff the positions of pattern P are identical in these paths. This also works for U_{k}F-equivalence classes. - Sergey Kirgizov, Apr 08 2018

			G.f. = 1 + x + x^2 + 2*x^3 + 4*x^4 + 7*x^5 + 13*x^6 + 26*x^7 + 52*x^8 + 104*x^9 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Andrei Asinowski, Cyril Banderier, and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).
Jean-Luc Baril, Sergey Kirgizov and Armen Petrossian, Enumeration of Łukasiewicz paths modulo some patterns, arXiv:1804.01293 [math.CO], 2018.
Paul Barry, Continued fractions and transformations of integer sequences, JIS 12 (2009) 09.7.6.
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018.
Paul Barry, Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials, arXiv:1910.00875 [math.CO], 2019.
Paul Barry, Elliptic Curves, Riordan arrays and Lattice Paths, arXiv:2507.16765 [math.CO], 2025. See p. 16.
Johann Cigler, Some nice Hankel determinants, arXiv preprint arXiv:1109.1449 [math.CO], 2011.
Shalosh B. Ekhad and Mingjia Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, (2017)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 666
Feihu Liu, Ying Wang, Yingrui Zhang, and Zihao Zhang, Hankel Determinants for Convolution of Power Series: An Extension of Cigler's Results, arXiv:2503.17187 [math.CO], 2025. See p. 9.
K. Park and G.-S. Cheon, Lattice path counting with a bounded strip restriction
Helmut Prodinger, Motzkin paths of bounded height with two forbidden contiguous subwords of length two, arXiv:2310.12497 [math.CO], 2023.

Cf. A000108, A001006, A004148, A006318, A025246.

Cf. A014137, A090344, A144700.

a023431 n = a023431_list !! n
a023431_list = 1 : 1 : f [1,1] where
   f xs'@(x:_:xs) = y : f (y : xs') where
     y = x + sum (zipWith (*) xs $ reverse $ xs')
-- Reinhard Zumkeller, Nov 13 2012

[(&+[Binomial(n-k, 2*k)*Catalan(k): k in [0..Floor(n/3)]]): n in [0..40]]; // G. C. Greubel, Jun 15 2022

a := n -> hypergeom([1/3 - n/3, 2/3 - n/3, -n/3], [2, -n], 27):
seq(simplify(a(n)), n = 0..32); # Peter Luschny, Jun 15 2022

a[0]=1; a[n_]:= a[n]= a[n-1] + Sum[a[k]*a[n-3-k], {k, 0, n-3}];
Table[a[n], {n,0,40}]

{a(n) = polcoeff( (1 - x - sqrt((1-x)^2 - 4*x^3 + x^4 * O(x^n))) / 2, n+3)}; /* Michael Somos, Jul 13 2003 */

[sum(binomial(n-k,2*k)*catalan_number(k) for k in (0..(n//3))) for n in (0..40)] # G. C. Greubel, Jun 15 2022

G.f.: (1 - x - sqrt((1-x)^2 - 4*x^3)) / (2*x^3) = A(x). y = x * A(x) satisfies 0 = x - y + x*y + (x*y)^2. - Michael Somos, Jul 13 2003
a(n+1) = a(n) + a(0)*a(n-2) + a(1)*a(n-3) + ... + a(n-2)*a(0). - Michael Somos, Jul 13 2003
a(n) = A025246(n+3). - Michael Somos, Jan 20 2004
G.f.: (1/(1-x))*c(x^3/(1-x)^2), c(x) the g.f. of A000108. - From Paul Barry, Sep 19 2008
From Paul Barry, May 22 2009: (Start)
G.f.: 1/(1-x-x^3/(1-x-x^3/(1-x-x^3/(1-x-x^3/(1-... (continued fraction).
a(n) = Sum_{k=0..floor(n/3)} binomial(n-k, 2*k)*A000108(k). (End)
(n+3)*a(n) = (2*n+3)*a(n-1) - n*a(n-2) + 2*(2*n-3)*a(n-3). - R. J. Mathar, Nov 26 2012
0 = a(n)*(16*a(n+1) - 10*a(n+2) + 32*a(n+3) - 22*a(n+4)) + a(n+1)*(2*a(n+1) - 15*a(n+2) + 9*a(n+3) + 4*a(n+4)) + a(n+2)*(a(n+2) + 2*a(n+3) - 5*a(n+4)) + a(n+3)*(a(n+3) + a(n+4)) if n>=0. - Michael Somos, Jan 30 2014
a(n) ~ (8 + 12*r^2 + 5*r) * sqrt(r^2 - 4*r + 3) / (4 * sqrt(Pi) * n^(3/2) * r^n), where r = 0.432040800333095789... is the real root of the equation -1 + 2*r - r^2 + 4*r^3 = 0. - Vaclav Kotesovec, Jun 15 2022
a(n) = hypergeom([(1 - n)/3, (2 - n)/3, -n/3], [2, -n], 27). - Peter Luschny, Jun 15 2022

A004149 Generalized Catalan numbers: a(n+1) = a(n) + Sum_{k=2..n-1} a(k)*a(n-1-k).

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 8, 16, 33, 69, 146, 312, 673, 1463, 3202, 7050, 15605, 34705, 77511, 173779, 390966, 882376, 1997211, 4532593, 10311720, 23512376, 53724350, 122995968, 282096693, 648097855, 1491322824, 3436755328, 7931085771

Offset: 0

N. J. A. Sloane

Number of Motzkin paths of length n-1 (n>=1) with no peaks and no valleys, i.e., no UD's and no DU's, where U=(1,1) and D=(1,-1). Example: a(7)=16 because there are 17 peakless Motzkin paths of length 6 (see A004148) of which only UHDUHD has a valley (here H=(1,0)). - Emeric Deutsch, Jan 08 2004
a(n+2) = number of Motzkin n-paths that avoid both UU and DD = number of Motzkin n-paths that avoid both UU and UFU. Example: a(7)=16 since of the 21 Motzkin 5-paths, only FUUDD, UFUDD, UUDDF, UUDFD, UUFDD contain a UU or DD (or both). Likewise, only FUUDD, UFUDD, UUDDF, UUDFD, UUFDD contain a UU or UFU. - David Callan, Jul 15 2004
Number of peakless Motzkin paths of length n without UHD's; here U=(1,1), H=(1,0), and D=(1,-1). Example: a(4)=2 because we have HHHH and UHHD.
a(n) = A190172(n,0).

			G.f. = 1 + x + x^2 + x^3 + 2*x^4 + 4*x^5 + 8*x^6 + 16*x^7 + 33*x^8 + 69*x^9 + ...

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

Seiichi Manyama, Table of n, a(n) for n = 0..2626 (first 201 terms from T. D. Noe)
Andrei Asinowski, Cyril Banderier, and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).
Paul Barry, Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials, arXiv:1910.00875 [math.CO], 2019.
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210; arXiv:math/0205301 [math.CO], 2002.
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
Tomislav Došlić, Dragutin Svrtan, and Darko Veljan, Enumerative aspects of secondary structures, Discr. Math., 285 (2004), 67-82.
Shanzhen Gao and Keh-Hsun Chen, Tackling Sequences From Prudent Self-Avoiding Walks, FCS'14, The 2014 International Conference on Foundations of Computer Science (Sequence 8 mentions a g.f. that gives a sequence that is similar to this sequence but without the first term).
Emanuele Munarini, Combinatorial properties of the antichains of a garland, Integers, 9 (2009), 353-374.
Helmut Prodinger, Motzkin paths of bounded height with two forbidden contiguous subwords of length two, arXiv:2310.12497 [math.CO], 2023.
Helmut Prodinger, Cornerless, peakless, valleyless Motzkin paths (regular and skew) and applications to bargraphs, arXiv:2501.13645 [math.CO], 2025. See p. 8.
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]
E. J. Janse van Rensburg, Adsorbing bargraph paths in a q-wedge, Journal of Physics A, v.38 n.40, 8505-8525.
M. S. Waterman, Home Page (contains copies of his papers)
Yan Zhuang, A generalized Goulden-Jackson cluster method and lattice path enumeration, Discrete Mathematics 341.2 (2018): 358-379; arXiv:1508.02793 [math.CO], 2015-2018.

Third row of A064645.

Cf. A001006, A004148.

a004149 n = a004149_list !! n
a004149_list = 1 : 1 : 1 : f [1,1,1] where
   f xs = y : f (y : xs) where
     y = head xs + sum (zipWith (*) (init $ init $ tail xs) (reverse xs))
-- Reinhard Zumkeller, Nov 13 2012

For Maple code producing the g.f. see A004148.
# Alternative:
p:= gfun:-rectoproc({(n-1)*a(n)+(2*n+1)*a(n+1)+(-n-2)*a(n+2)+(-5-n)*a(n+4)+(-13-2*n)*a(n+5)+(n+8)*a(n+6), a(0) = 1, a(1) = 1, a(2) = 1, a(3) = 1, a(4) = 2, a(5) = 4},a(n),remember):
map(p, [$0..100]); # Robert Israel, May 07 2015

a[ 0 ]=1; a[ n_Integer ] := a[ n ]=a[ n-1 ]+Sum[ a[ k ]*a[ n-2-k ], {k, 2, n-2} ];
CoefficientList[Series[2/(1-x+x^2+x^3+Sqrt[(1-x^4)(1-2x-x^2)]),{x,0,40}],x] (* Harvey P. Dale, Aug 09 2017 *)

{a(n) = polcoeff( (1 - x + x^2 + x^3 - sqrt( (1 - x^4) * (1 - 2*x - x^2) + x^3 * O(x^n))) / (2*x^2), n)}; /* Michael Somos, Oct 28 2005 */

G.f.: 2/(1 - z + z^2 + z^3 + sqrt((1-z^4)(1-2z-z^2))). - Emeric Deutsch, Jan 08 2004
G.f.: 1/(1-x-x^4/(1-x-x^2-x^3-x^4/(1-x-x^2-x^3-x^4/(1-... (continued fraction). - Paul Barry, May 22 2009
D-finite with recurrence: (n+2)*a(n) = (2*n+1)*a(n-1) + (n-1)*a(n-2) + (n-4)*a(n-4) - (2*n-11)*a(n-5) - (n-7)*a(n-6). - Vaclav Kotesovec, Aug 10 2013
a(n) ~ (1+sqrt(2))^(n+1/2)/(sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 10 2013
G.f. g(x) satisfies x^2*g^2 - (1-x+x^2+x^3)*g + 1 = 0 and
(x^4-1)*(x^2+2*x-1)*x*g'(x) - (x^3-x+2)*(x^3+x^2+x-1)*g(x) + 4*x^3+2*x^2-2 = 0. - Robert Israel, May 07 2015
0 = a(n)*(+a(n+1) + 5*a(n+2) - 4*a(n+3) - 7*a(n+5) - 17*a(n+6) + 10*a(n+7)) + a(n+1)*(-a(n+1) + 6*a(n+2) - 5*a(n+3) + 5*a(n+4) + 2*a(n+5) - 36*a(n+6) + 17*a(n+7)) + a(n+2)*(+a(n+2) + a(n+3) + 7*a(n+4) + 24*a(n+5) - 2*a(n+6) - 7*a(n+7)) + a(n+3)*(-2*a(n+4) - 7*a(n+5) + 5*a(n+6)) + a(n+4)*(+a(n+5) + 5*a(n+6) - 4*a(n+7)) + a(n+5)*(-a(n+5) + 6*a(n+6) - 5*a(n+7)) + a(n+6)*(+a(n+6) + a(n+7)) for all n>=0. - Michael Somos, Jan 09 2017
G.f.: 1/G(x), with G(x)=1-(x-x^3)/(1-x^2/G(x)) (continued fraction). - Nikolaos Pantelidis, Jan 11 2023

A023426 a(n) = a(n-1) + Sum_{k=0..n-4} a(k)*a(n-4-k), a(0) = 1. Generalized Catalan Numbers.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 7, 11, 18, 32, 59, 107, 191, 343, 627, 1159, 2146, 3972, 7373, 13757, 25781, 48437, 91165, 171945, 325096, 616066, 1169667, 2224355, 4236728, 8082374, 15441719, 29542411, 56590472, 108532322, 208387711, 400551615, 770710831, 1484383399

Offset: 0

Olivier Gérard

Number of lattice paths from (0,0) to (n,0) that stay weakly in the first quadrant and such that each step is either U=(2,1),D=(2,-1), or H=(1,0). E.g. a(5)=4 because we have HHHHH, HUD, UDH and UHD. - Emeric Deutsch, Dec 23 2003

Hankel transform is A132380(n+3). - Paul Barry, May 22 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Andrei Asinowski, Cyril Banderier and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See pp. 10, 19-21.
K. Park and G.S. Cheon, Lattice path counting with a bounded strip restriction

Cf. A000108, A001006, A004148, A006318.

a := n -> hypergeom([(1 - n)/4, (2 - n)/4, (3 - n)/4, -n/4], [2, (1 - n)/2, -n/2], 2^6): seq(simplify(a(n)), n = 0..35); # Peter Luschny, Jul 12 2024

Clear[ a ]; a[ 0 ]=1; a[ n_Integer ] := a[ n ]=a[ n-1 ]+Sum[ a[ k ]*a[ n-4-k ], {k, 0, n-4} ];
CoefficientList[Series[(1-x-Sqrt[(1-x)^2-4*x^4])/(2*x^4), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 01 2014 *)

G.f.: [1-z-sqrt((1-z)^2-4z^4)]/[2z^4]. - Emeric Deutsch, Dec 23 2003
From Paul Barry, May 22 2009: (Start)
G.f.: 1/(1-x-x^4/(1-x-x^4/(1-x-x^4/(1-x-x^4/(1-... (continued fraction).
G.f.: (1/(1-x))c(x^4/(1-x)^2), c(x) the g.f. of A000108.
a(n) = Sum_{k=0..floor(n/4)} C(n-2k,2k)*A000108(k). (End)
D-finite with recurrence (n+4)*a(n) +(n+4)*a(n-1) -(5*n+8)*a(n-2) +3*n*a(n-3) +4*(2-n)*a(n-4) +12*(3-n)*a(n-5)=0. - R. J. Mathar, Sep 29 2012
a(n) ~ sqrt(3) * 2^(n+3/2) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 01 2014
G.f. A(x) satisfies: A(x) = (1 + x^4 * A(x)^2) / (1 - x). - Ilya Gutkovskiy, Jul 20 2021
a(n) = hypergeom([(1 - n)/4, (2 - n)/4, (3 - n)/4, -n/4], [2, (1 - n)/2, -n/2], 64). - Peter Luschny, Jul 12 2024

Name extended by a formula from the author in Mathematica by Peter Luschny, Jul 13 2024

A003600 Maximal number of pieces obtained by slicing a torus (or a bagel) with n cuts: (n^3 + 3n^2 + 8n)/6 (n > 0).

Original entry on oeis.org

1, 2, 6, 13, 24, 40, 62, 91, 128, 174, 230, 297, 376, 468, 574, 695, 832, 986, 1158, 1349, 1560, 1792, 2046, 2323, 2624, 2950, 3302, 3681, 4088, 4524, 4990, 5487, 6016, 6578, 7174, 7805, 8472, 9176, 9918, 10699, 11520, 12382, 13286, 14233, 15224

Offset: 0

N. J. A. Sloane, Mira Bernstein

Both the bagel and the torus are solid (apart from the hole in the middle, of course)! - N. J. A. Sloane, Oct 03 2012

M. Gardner, The 2nd Scientific American Book of Mathematical Puzzles and Diversions. Simon and Schuster, NY, 1961. See Chapter 13. (See pages 113-116 in the English edition published by Pelican Books in 1966.)
Clifford A. Pickover, Computers and the Imagination, St. Martin's Press, NY, 1991, pp. 373-374 and Plate 27.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
George Hart, Slice a Bagel into 13 Pieces with Three Cuts
Kyu-Hwan Lee and Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.
Clifford A. Pickover, Illustration of a(3)=13 [Plate 27 from Computers and the Imagination, used with permission]
N. J. A. Sloane, Illustration for a(2)=6 and a(3)=13 [Based on part of Fig. 62 in M. Gardner, The 2nd Scientific American Book of Mathematical Puzzles and Diversions, colored and annotated]
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 1.
Eric Weisstein's World of Mathematics, Torus Cutting.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000124 (slicing a pancake), A000125 (a cake).

Cf. A004148.

I:=[1, 2, 6, 13, 24]; [n le 5 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jun 29 2012

CoefficientList[Series[(1-2*x+4*x^2-3*x^3+x^4)/(1-x)^4,{x,0,40}],x] (* Vincenzo Librandi, Jun 29 2012 *)
LinearRecurrence[{4,-6,4,-1},{1,2,6,13,24},50] (* Harvey P. Dale, Oct 22 2016 *)

a(n)=if(n,n*(n^2+3*n+8)/6,1) \\ Charles R Greathouse IV, Oct 07 2015

a(n) = binomial(n+2, n-1) + binomial(n, n-1).
a(n) = coefficient of z^3 in the series expansion of G^n (n>0), where G=[1-z+z^2-sqrt(1-2z-z^2-2z^3+z^4)]/(2z^2) is the g.f. of A004148 (secondary structures of RNA molecules). - Emeric Deutsch, Jan 11 2004
Binomial transform of [1, 1, 3, 0, 1, -1, 1, -1, 1, ...]. - Gary W. Adamson, Nov 08 2007
G.f.: (1 - 2*x + 4*x^2 - 3*x^3 + x^4) / (1 - x)^4. - Colin Barker, Jun 28 2012
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 29 2012
a(n) = A108561(n+4,3) for n > 0. - Reinhard Zumkeller, Jun 10 2005
a(n) = A000292(n+1) - A000124(n) for n > 0. - Torlach Rush, Aug 04 2018
a(n) = A000125(n+1) - 2, as one can see by thinking of the donut hole as a slit in a cake, i.e. an (n+1)st cut in the cake that doesn't quite reach the edges of the cake and so leaves two pieces unseparated. - Glen Whitney, Mar 31 2019
E.g.f.: 1 + exp(x)*x*(12 + 6*x + x^2)/6. - Stefano Spezia, Apr 19 2025

More terms from James Sellers, Aug 22 2000

A023427 Number of Dyck n-paths with ascents and descents of length equal to 1 (mod 4).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 4, 7, 11, 17, 28, 49, 87, 152, 262, 453, 794, 1408, 2507, 4462, 7943, 14179, 25415, 45713, 82398, 148731, 268859, 486890, 883411, 1605582, 2922259, 5325377, 9716564, 17750332, 32464980, 59443403, 108951953, 199886003, 367052947, 674620772

Offset: 0

Olivier Gérard

Number of secondary structures of size n having no stacks of odd length (see Hofacker et al., p. 209). - Emeric Deutsch, Dec 26 2011

a(n) is the number of Dyck n-paths all of whose ascents and descents have lengths equal to 1 (mod 4). The a(5) = 2 paths for n=5 are: UDUDUDUDUD, UUUUUDDDDD. - Alois P. Heinz, May 09 2012

			(5)=2; representing unpaired vertices by v and arcs by AA, BB, etc., the 8 (= A004148(5)) secondary structures of size 5 are vvvvv, AvAvv, vvAvA, AvvAv, vAvvA, AvvvA, vAvAv, ABvBA; they have 0,1,1,1,1,1,1,0 stacks of odd length, respectively. - _Emeric Deutsch_, Dec 26 2011

Alois P. Heinz, Table of n, a(n) for n = 0..1000
I. L. Hofacker, P. Schuster and P. F. Stadler, Combinatorics of RNA secondary structures, Discrete Appl. Math., 88, 1998, 207-237.
P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26 (1979), 261-272.

Cf. A000108, A001006, A004148, A006318, A216116, A202845 - A202849.

f := z^4/(1-z^4): eq := G = 1+z*G+f*G*(G-1)/(1+f): G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 42)): seq(coeff(Gser, z, n), n = 0 .. 39); # Emeric Deutsch, Dec 26 2011
a:= proc(n) option remember;
      `if`(n=0, 1, a(n-1) +add(a(k)*a(n-4-k), k=1..n-4))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, May 09 2012

Clear[ a ]; a[ 0 ]=1; a[ n_Integer ] := a[ n ]=a[ n-1 ]+Sum[ a[ k ]*a[ n-4-k ], {k, 1, n-4} ];
CoefficientList[Series[((1-x+x^4) - Sqrt[(1-x+x^4)^2 - 4*x^4])/(2*x^4), {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 16 2013 *)

a(n):=if n=0 then 1 else sum(binomial(n-3*q,((q)))*binomial((n-3*q),(q+1))/(n-3*q),q,0,(n-1)/3); /* Vladimir Kruchinin, Jan 21 2019 */

{a(n)=polcoeff(((1-x+x^4) - sqrt((1-x+x^4)^2 - 4*x^4 +x^5*O(x^n)))/(2*x^4), n)} \\ Paul D. Hanna, Oct 29 2012

{a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1 + x*A/(1-x^4*A+x*O(x^n))); polcoeff(A, n)} \\ Paul D. Hanna, Oct 29 2012

a(n) = A202845(n,0). A(x) satisfies A=1+x*A+f*A*(A-1)/(1+f), where f=x^4/(1-x^4). - Emeric Deutsch, Dec 26 2011
G.f.: A(x) = ((1-x+x^4) - sqrt((1-x+x^4)^2 - 4*x^4))/(2*x^4). - Paul D. Hanna, Oct 29 2012
G.f. satisfies: A(x) = 1 + x*A(x)/(1 - x^4*A(x)). - Paul D. Hanna, Oct 29 2012
G.f.: 1 + x*exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} C(n,k)^2 * x^(3*k) ). - Paul D. Hanna, Oct 29 2012
a(n) = A216116(n-1) for n>0.
Recurrence: (n+4)*a(n) = (2*n+5)*a(n-1) - (n+1)*a(n-2) + 2*(n-2)*a(n-4) + (2*n-7)*a(n-5) - (n-8)*a(n-8). - Vaclav Kotesovec, Sep 16 2013
a(n) ~ sqrt(-4*c^2-3*c^3+4-4*c)*(1+2*c-c^3)^n*(-5*c^3-3*c^2+9*c+10) / (2*n^(3/2)*sqrt(Pi)), where c = 1/2*sqrt((4+(155/2-3*sqrt(849)/2)^(1/3) +(155/2+3*sqrt(849)/2)^(1/3))/3) - 1/2*sqrt(8/3-1/3*(155/2-3*sqrt(849)/2)^(1/3) - 1/3*(155/2+3*sqrt(849)/2)^(1/3) + 2*sqrt(3/(4+(155/2-3*sqrt(849)/2)^(1/3) + (155/2+3*sqrt(849)/2)^(1/3)))) = 0.5248885986564... is the root of the equation c^4-2*c^2-c+1=0. - Vaclav Kotesovec, Sep 16 2013
a(n) = Sum_{k=0..(n-1)/3} C(n-3*k,k)*C(n-3*k,k+1)/(n-3*k), n>0, a(0)=1. - Vladimir Kruchinin, Jan 21 2019

New name, using a comment of Alois P. Heinz, from Peter Luschny, Jan 21 2019

A200074 G.f. satisfies A(x) = (1 + xA(x)^2)(1 + x^2*A(x)).

Original entry on oeis.org

1, 1, 3, 9, 30, 108, 406, 1577, 6280, 25499, 105169, 439388, 1855636, 7908909, 33975250, 146954693, 639460707, 2797384235, 12295494109, 54272825103, 240480529815, 1069257987503, 4769306203838, 21334400243252, 95687482105807, 430217846136134, 1938651904470374, 8754225470415889

Offset: 0

Paul D. Hanna, Nov 13 2011

nonn

More generally, for fixed parameters p, q, r, and s, if F(x) satisfies:
F(x) = exp( Sum_{n>=1} x^(n*r)*F(x)^(n*p)/n * [Sum_{k=0..n} C(n,k)^2 * x^(k*s)*F(x)^(k*q)] ),
then F(x) = (1 + x^r*F(x)^(p+1))*(1 + x^(r+s)*F(x)^(p+q+1)).

			G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 30*x^4 + 108*x^5 + 406*x^6 + 1577*x^7 +...
Related expansions:
A(x)^2 = 1 + 2*x + 7*x^2 + 24*x^3 + 87*x^4 + 330*x^5 + 1289*x^6 +...
A(x)^3 = 1 + 3*x + 12*x^2 + 46*x^3 + 180*x^4 + 720*x^5 + 2928*x^6 +...
where A(x) = 1 + x*A(x)^2 + x^2*A(x) + x^3*A(x)^3.
The logarithm of the g.f. A = A(x) equals the series:
log(A(x)) = (A + x)*x + (A^2 + 2^2*x*A + x^2)*x^2/2 +
(A^3 + 3^2*x*A^2 + 3^2*x^2*A + x^3)*x^3/3 +
(A^4 + 4^2*x*A^3 + 6^2*x^2*A^2 + 4^2*x^3*A + x^4)*x^4/4 +
(A^5 + 5^2*x*A^4 + 10^2*x^2*A^3 + 10^2*x^3*A^2 + 5^2*x^4*A + x^5)*x^5/5 +
(A^6 + 6^2*x*A^5 + 15^2*x^2*A^4 + 20^2*x^3*A^3 + 15^2*x^4*A^2 + 6^2*x^5*A + x^6)*x^6/6 +...
more explicitly,
log(A(x)) = x + 5*x^2/2 + 19*x^3/3 + 77*x^4/4 + 331*x^5/5 + 1445*x^6/6 + 6392*x^7/7 + 28565*x^8/8 +...

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A199876, A199877, A198951, A198953, A198957, A192415, A198888, A004148, A036765.

Cf. A186241, A199874, A200075, A200718, A200719, A215576.

a:= n-> coeff(series(RootOf(A=(1+x*A^2)*(1+x^2*A), A), x, n+1), x, n):
seq(a(n), n=0..30);  # Alois P. Heinz, May 16 2012

m = 28; A[_] = 0;
Do[A[x_] = (1 + x A[x]^2)(1 + x^2 A[x]) + O[x]^m, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Oct 02 2019 *)

{a(n)=local(A=1+x);for(i=1,n,A=(1+x*A^2)*(1+x^2*A^1)+x*O(x^n));polcoeff(A,n)}

{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2*x^j/A^j)*(x*A+x*O(x^n))^m/m))); polcoeff(A, n, x)}

{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, (1-x/A)^(2*m+1)*sum(j=0, n, binomial(m+j, j)^2*x^j/A^j)*x^m*A^m/m))); polcoeff(A, n, x)}

G.f. satisfies:
(1) A(x) = exp( Sum_{n>=1} (Sum_{k=0..n} C(n,k)^2 * x^k*A(x)^(n-k)) * x^n/n ).
(2) A(x) = exp( Sum_{n>=1} (1-x/A(x))^(2*n+1)*(Sum_{k>=0} C(n+k,k)^2*x^k/A(x)^k) * x^n*A(x)^n/n ).
(3) A(x) = x / Series_Reversion( x*G(x) ) where G(x) is the g.f. of A199876.
(4) A(x) = G(x/A(x)) where G(x) = A(x*G(x)) is the g.f. of A199876.
Recurrence: (n+1)*(n+2)*(1241*n^4 - 10636*n^3 + 25417*n^2 - 7382*n - 17136)*a(n) =  - 18*(n+1)*(443*n^3 - 3889*n^2 + 9734*n - 5712)*a(n-1) + 4*(6205*n^6 - 53180*n^5 + 115741*n^4 + 64762*n^3 - 370103*n^2 + 246727*n - 25704)*a(n-2) + 6*(2482*n^6 - 24995*n^5 + 76519*n^4 - 36347*n^3 - 185471*n^2 + 293092*n - 140400)*a(n-3) + 2*(4964*n^6 - 57436*n^5 + 228617*n^4 - 276802*n^3 - 361447*n^2 + 956696*n - 320496)*a(n-4) - 6*(2482*n^6 - 32441*n^5 + 140587*n^4 - 173153*n^3 - 266705*n^2 + 677518*n - 291840)*a(n-5) + 12*(n-4)*(2*n - 11)*(11*n^2 + 73*n - 748)*a(n-6) + 2*(n-5)*(2*n - 13)*(1241*n^4 - 5672*n^3 + 955*n^2 + 16508*n - 8496)*a(n-7). - Vaclav Kotesovec, Aug 18 2013
a(n) ~ c*d^n/n^(3/2), where d = 4.770539985405... is the root of the equation -4 + 12*d^2 - 8*d^3 - 12*d^4 - 20*d^5 + d^7 = 0 and c = 0.612892860188927397373456... - Vaclav Kotesovec, Aug 18 2013
a(n) = Sum_{k=0..floor(n/2)} binomial(2*n-3*k+1,k) * binomial(2*n-3*k+1,n-2*k) / (2*n-3*k+1). - Seiichi Manyama, Jul 18 2023

A200075 G.f. satisfies A(x) = (1 + xA(x)^2)(1 + x^2*A(x)^3).

Original entry on oeis.org

1, 1, 3, 11, 45, 198, 914, 4367, 21414, 107155, 544987, 2808978, 14640073, 77025373, 408544815, 2182206259, 11727989593, 63373962690, 344109933186, 1876562458845, 10273572074493, 56443282489240, 311097732946200, 1719707775782826, 9531914043637385, 52963938340248863, 294966593345731623

Offset: 0

Paul D. Hanna, Nov 13 2011

nonn

More generally, for fixed parameters p, q, r, and s, if F(x) satisfies:
F(x) = exp( Sum_{n>=1} x^(n*r)*F(x)^(n*p)/n * [Sum_{k=0..n} C(n,k)^2 * x^(k*s)*F(x)^(k*q)] ),
then F(x) = (1 + x^r*F(x)^(p+1))*(1 + x^(r+s)*F(x)^(p+q+1)).

			G.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 45*x^4 + 198*x^5 + 914*x^6 +...
Related expansions:
A(x)^2 = 1 + 2*x + 7*x^2 + 28*x^3 + 121*x^4 + 552*x^5 + 2615*x^6 +...
A(x)^3 = 1 + 3*x + 12*x^2 + 52*x^3 + 237*x^4 + 1122*x^5 + 5463*x^6 +...
A(x)^5 = 1 + 5*x + 25*x^2 + 125*x^3 + 630*x^4 + 3211*x^5 + 16545*x^6 +...
where A(x) = 1 + x*A(x)^2 + x^2*A(x)^3 + x^3*A(x)^5.
The logarithm of the g.f. A = A(x) equals the series:
log(A(x)) = (1 + x*A)*x*A + (1 + 2^2*x*A + x^2*A^2)*x^2*A^2/2 +
(1 + 3^2*x*A + 3^2*x^2*A^2 + x^3*A^3)*x^3*A^3/3 +
(1 + 4^2*x*A + 6^2*x^2*A^2 + 4^2*x^3*A^3 + x^4*A^4)*x^4*A^4/4 +
(1 + 5^2*x*A + 10^2*x^2*A^2 + 10^2*x^3*A^3 + 5^2*x^4*A^4 + x^5*A^5)*x^5*A^5/5 +
(1 + 6^2*x*A + 15^2*x^2*A^2 + 20^2*x^3*A^3 + 15^2*x^4*A^4 + 6^2*x^5*A^5 + x^6*A^6)*x^6*A^6/6 +...
more explicitly,
log(A(x)) = x + 5*x^2/2 + 25*x^3/3 + 129*x^4/4 + 686*x^5/5 + 3713*x^6/6 + 20350*x^7/7 +...
Given G(x) where 1+x*G(x) is the g.f. of A004148, then the coefficients in the powers of G(x) begin:
1: [(1), 1, 2, 4, 8, 17, 37, 82, 185, 423, 978, ...];
2: [1,(2), 5, 12, 28, 66, 156, 370, 882, 2112, ...];
3: [1, 3,(9), 25, 66, 171, 437, 1107, 2790, 7009, ...];
4: [1, 4, 14,(44), 129, 364, 1000, 2696, 7172, 18892, ...];
5: [1, 5, 20, 70,(225), 686, 2015, 5760, 16135, 44500, ...];
6: [1, 6, 27, 104, 363,(1188), 3713, 11214, 32994, 95106, ...];
7: [1, 7, 35, 147, 553, 1932,(6398), 20350, 62734, 188650, ...];
8: [1, 8, 44, 200, 806, 2992, 10460,(34936), 112585, 352560, ...];
9: [1, 9, 54, 264, 1134, 4455, 16389, 57330,(192726), 627406, ...]; ...;
the coefficients in parenthesis form the initial terms of this sequence:
[1/1, 2/2, 9/3, 44/4, 225/5, 1188/6, 6398/7, 34936/8, 192726/9, ...].
The coefficients in the logarithm of the g.f. is also a diagonal in the above table.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A004148, A199876, A199877, A198951, A198953, A198957, A192415, A198888, A036765.

Cf. A186241, A199874, A200074, A200718, A200719, A215576.

CoefficientList[1/x*InverseSeries[Series[x*(1-x-x^2 + Sqrt[(1+x+x^2)*(1-3*x+x^2)])/2,{x,0,20}],x],x] (* Vaclav Kotesovec, Sep 19 2013 *)

{a(n)=local(A=1+x);for(i=1,n,A=(1+x*A^2)*(1+x^2*A^3)+x*O(x^n));polcoeff(A,n)}

{a(n)=polcoeff(1/x*serreverse(x*(1-x-x^2 + sqrt((1+x+x^2)*(1-3*x+x^2)+x*O(x^n)))/2),n)}

{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2*x^j*A^j)*(x*A+x*O(x^n))^m/m))); polcoeff(A, n, x)}

{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, (1-x*A)^(2*m+1)*sum(j=0, n, binomial(m+j, j)^2*x^j*A^j)*x^m*A^m/m))); polcoeff(A, n, x)}

G.f.: (1/x)*Series_Reversion( x*(1-x-x^2 + sqrt((1+x+x^2)*(1-3*x+x^2)))/2 ).
a(n) = [x^n] G(x)^(n+1)/(n+1), where 1+x*G(x) is the g.f. of A004148.
G.f. A(x) satisfies:
(1) A(x) = (1/x)*Series_Reversion( x/G(x) ) where 1+x*G(x) is the g.f. of A004148.
(2) A(x) = G(x*A(x)) where G(x) = A(x/G(x)) and 1+x*G(x) is the g.f. of A004148.
(3) A(x) = exp( Sum_{n>=1} (Sum_{k=0..n} C(n,k)^2 * x^k*A(x)^k) * x^n*A(x)^n/n ).
(4) A(x) = exp( Sum_{n>=1} (1-x*A(x))^(2*n+1)*(Sum_{k>=0} C(n+k,k)^2*x^k*A(x)^k) * x^n*A(x)^n/n ).
Recurrence: 8*n*(2*n+1)*(4*n+1)*(4*n+3)*(1557671*n^7 - 18939961*n^6 + 94817789*n^5 - 252067387*n^4 + 381880748*n^3 - 327052012*n^2 + 145198992*n - 25583040)*a(n) = (2026529971*n^11 - 24640889261*n^10 + 122927623620*n^9 - 322351865586*n^8 + 467303512311*n^7 - 343677276405*n^6 + 61590777290*n^5 + 76066203476*n^4 - 45605627832*n^3 + 4625651136*n^2 + 1916801280*n - 338688000)*a(n-1) + 2*(800642894*n^11 - 10936104295*n^10 + 62803409541*n^9 - 196202081616*n^8 + 357730085364*n^7 - 370711524567*n^6 + 174415015309*n^5 + 25877389846*n^4 - 63266190708*n^3 + 19055552472*n^2 + 1313789760*n - 861840000)*a(n-2) + 6*(308418858*n^11 - 4675368852*n^10 + 30103912361*n^9 - 106665982366*n^8 + 223860428776*n^7 - 274000455628*n^6 + 166116940489*n^5 - 2432493994*n^4 - 54297743044*n^3 + 22033617000*n^2 + 936446400*n - 1315440000)*a(n-3) + 6*(n-2)*(2*n-7)*(3*n-10)*(3*n-8)*(1557671*n^7 - 8036264*n^6 + 13889114*n^5 - 7559372*n^4 - 2491645*n^3 + 2975476*n^2 - 179460*n - 187200)*a(n-4). - Vaclav Kotesovec, Sep 19 2013
a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = 1301/1024 + 1/(1024*sqrt(3/(7183147 - (2002819072*2^(2/3))/(3725055779 + 42057117*sqrt(16305))^(1/3) + 1024*(7450111558 + 84114234*sqrt(16305))^(1/3)))) + (1/2)*sqrt(7183147/393216 - (3725055779 + 42057117*sqrt(16305))^(1/3)/(384*2^(2/3)) + 977939/(192*(7450111558 + 84114234*sqrt(16305))^(1/3)) + (1/131072)*(4194454317*sqrt(3/(7183147 - (2002819072*2^(2/3))/(3725055779 + 42057117*sqrt(16305))^(1/3) + 1024*(7450111558 + 84114234*sqrt(16305))^(1/3))))) = 5.89828930084513611... is the root of the equation -108 - 1188*d - 1028*d^2 - 1301*d^3 + 256*d^4 = 0 and c = 0.656947859044624009263362998790812821830934... - Vaclav Kotesovec, Sep 19 2013
a(n) = Sum_{k=0..floor(n/2)} binomial(2*n-k+1,k) * binomial(2*n-k+1,n-2*k) / (2*n-k+1). - Seiichi Manyama, Jul 18 2023

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cp's OEIS Frontend

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

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A005251 a(0) = 0, a(1) = a(2) = a(3) = 1; thereafter, a(n) = a(n-1) + a(n-2) + a(n-4).

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A110320 Number of blocks in all RNA secondary structures with n nodes (an RNA secondary structure can be viewed as a restricted noncrossing partition).

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A023431 Generalized Catalan Numbers x^3*A(x)^2 + (x-1)*A(x) + 1 =0.

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A004149 Generalized Catalan numbers: a(n+1) = a(n) + Sum_{k=2..n-1} a(k)*a(n-1-k).

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A023431 Generalized Catalan Numbers x^3A(x)^2 + (x-1)A(x) + 1 =0.

A003600 Maximal number of pieces obtained by slicing a torus (or a bagel) with n cuts: (n^3 + 3n^2 + 8n)/6 (n > 0).

A200074 G.f. satisfies A(x) = (1 + xA(x)^2)(1 + x^2*A(x)).

A200075 G.f. satisfies A(x) = (1 + xA(x)^2)(1 + x^2*A(x)^3).