This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A001764 M2926 N1174 #722 Aug 01 2025 07:06:41
%S A001764 1,1,3,12,55,273,1428,7752,43263,246675,1430715,8414640,50067108,
%T A001764 300830572,1822766520,11124755664,68328754959,422030545335,
%U A001764 2619631042665,16332922290300,102240109897695,642312451217745,4048514844039120,25594403741131680,162250238001816900
%N A001764 a(n) = binomial(3*n,n)/(2*n+1) (enumerates ternary trees and also noncrossing trees).
%C A001764 Smallest number of straight line crossing-free spanning trees on n points in the plane.
%C A001764 Number of dissections of some convex polygon by nonintersecting diagonals into polygons with an odd number of sides and having a total number of 2n+1 edges (sides and diagonals). - _Emeric Deutsch_, Mar 06 2002
%C A001764 Number of lattice paths of n East steps and 2n North steps from (0,0) to (n,2n) and lying weakly below the line y=2x. - _David Callan_, Mar 14 2004
%C A001764 With interpolated zeros, this has g.f. 2*sqrt(3)*sin(arcsin(3*sqrt(3)*x/2)/3)/(3*x) and a(n) = C(n+floor(n/2),floor(n/2))*C(floor(n/2),n-floor(n/2))/(n+1). This is the first column of the inverse of the Riordan array (1-x^2,x(1-x^2)) (essentially reversion of y-y^3). - _Paul Barry_, Feb 02 2005
%C A001764 Number of 12312-avoiding matchings on [2n].
%C A001764 Number of complete ternary trees with n internal nodes, or 3n edges.
%C A001764 Number of rooted plane trees with 2n edges, where every vertex has even outdegree ("even trees").
%C A001764 a(n) is the number of noncrossing partitions of [2n] with all blocks of even size. E.g.: a(2)=3 counts 12-34, 14-23, 1234. - _David Callan_, Mar 30 2007
%C A001764 Pfaff-Fuss-Catalan sequence C^{m}_n for m=3, see the Graham et al. reference, p. 347. eq. 7.66.
%C A001764 Also 3-Raney sequence, see the Graham et al. reference, p. 346-7.
%C A001764 The number of lattice paths from (0,0) to (2n,0) using an Up-step=(1,1) and a Down-step=(0,-2) and staying above the x-axis. E.g., a(2) = 3; UUUUDD, UUUDUD, UUDUUD. - Charles Moore (chamoore(AT)howard.edu), Jan 09 2008
%C A001764 a(n) is (conjecturally) the number of permutations of [n+1] that avoid the patterns 4-2-3-1 and 4-2-5-1-3 and end with an ascent. For example, a(4)=55 counts all 60 permutations of [5] that end with an ascent except 42315, 52314, 52413, 53412, all of which contain a 4-2-3-1 pattern and 42513. - _David Callan_, Jul 22 2008
%C A001764 Central terms of pendular triangle A167763. - _Philippe Deléham_, Nov 12 2009
%C A001764 With B(x,t)=x+t*x^3, the comp. inverse in x about 0 is A(x,t) = Sum_{j>=0} a(j) (-t)^j x^(2j+1). Let U(x,t)=(x-A(x,t))/t. Then DU(x,t)/Dt=dU/dt+U*dU/dx=0 and U(x,0)=x^3, i.e., U is a solution of the inviscid Burgers's, or Hopf, equation. Also U(x,t)=U(x-t*U(x,t),0) and dB(x,t)/dt = U(B(x,t),t) = x^3 = U(x,0). The characteristics for the Hopf equation are x(t) = x(0) + t*U(x(t),t) = x(0) + t*U(x(0),0) = x(0) + t*x(0)^3 = B(x(0),t). These results apply to all the Fuss-Catalan sequences with 3 replaced by n>0 and 2 by n-1 (e.g., A000108 with n=2 and A002293 with n=4), see also A086810, which can be generalized to A133437, for associahedra. - _Tom Copeland_, Feb 15 2014
%C A001764 Number of intervals (i.e., ordered pairs (x,y) such that x<=y) in the Kreweras lattice (noncrossing partitions ordered by refinement) of size n, see the Bernardi & Bonichon (2009) and Kreweras (1972) references. - _Noam Zeilberger_, Jun 01 2016
%C A001764 Number of sum-indecomposable (4231,42513)-avoiding permutations. Conjecturally, number of sum-indecomposable (2431,45231)-avoiding permutations. - _Alexander Burstein_, Oct 19 2017
%C A001764 a(n) is the number of topologically distinct endstates for the game Planted Brussels Sprouts on n vertices, see Ji and Propp link. - _Caleb Ji_, May 14 2018
%C A001764 Number of complete quadrillages of 2n+2-gons. See Baryshnikov p. 12. See also Nov 10 2014 comments in A134264. - _Tom Copeland_, Jun 04 2018
%C A001764 a(n) is the number of 2-regular words on the alphabet [n] that avoid the patterns 231 and 221. Equivalently, this is the number of 2-regular tortoise-sortable words on the alphabet [n] (see the Defant and Kravitz link). - _Colin Defant_, Sep 26 2018
%C A001764 a(n) is the number of Motzkin paths of length 3n with n steps of each type, with the condition that (1, 0) and (1, 1) steps alternate (starting with (1, 0)). - _Helmut Prodinger_, Apr 08 2019
%C A001764 a(n) is the number of uniquely sorted permutations of length 2n+1 that avoid the patterns 312 and 1342. - _Colin Defant_, Jun 08 2019
%C A001764 The compositional inverse o.g.f. pair in Copeland's comment above are related to a pair of quantum fields in Balduf's thesis by Theorem 4.2 on p. 92. - _Tom Copeland_, Dec 13 2019
%C A001764 The sequences of Fuss-Catalan numbers, of which this is the first after the Catalan numbers A000108 (the next is A002293), appear in articles on random matrices and quantum physics. See Banica et al., Collins et al., and Mlotkowski et al. Interpretations of these sequences in terms of the cardinality of specific sets of noncrossing partitions are provided by A134264. - _Tom Copeland_, Dec 21 2019
%C A001764 Call C(p, [alpha], g) the number of partitions of a cyclically ordered set with p elements, of cyclic type [alpha], and of genus g (the genus g Faa di Bruno coefficients of type [alpha]). This sequence counts the genus 0 partitions (non-crossing, or planar, partitions) of p = 3n into n parts of length 3: a(n) = C(3n, [3^n], 0). For genus 1 see A371250, for genus 2 see A371251. - _Robert Coquereaux_, Mar 16 2024
%C A001764 a(n) is the total number of down steps before the first up step in all 2_1-Dyck paths of length 3*n for n > 0. A 2_1-Dyck path is a lattice path with steps (1,2), (1,-1) that starts and ends at y = 0 and does not go below the line y = -1. - _Sarah Selkirk_, May 10 2020
%C A001764 a(n) is the number of pairs (A<=B) of noncrossing partitions of [n]. - _Francesca Aicardi_, May 28 2022
%C A001764 a(n) is the number of parking functions of size n avoiding the patterns 231 and 321. - _Lara Pudwell_, Apr 10 2023
%C A001764 Number of rooted polyominoes composed of n square cells of the hyperbolic regular tiling with Schläfli symbol {4,oo}. A rooted polyomino has one external edge identified, and chiral pairs are counted as two. A stereographic projection of the {4,oo} tiling on the Poincaré disk can be obtained via the Christensson link. - _Robert A. Russell_, Jan 27 2024
%C A001764 This is instance k = 3 of the family {C(k, n)}_{n>=0} given in a comment in A130564. - _Wolfdieter Lang_, Feb 05 2024
%C A001764 The number of Apollonian networks (planar 3-trees) with n+3 vertices with a given base triangle. - _Allan Bickle_, Feb 20 2024
%C A001764 Number of rooted polyominoes composed of n tetrahedral cells of the hyperbolic regular tiling with Schläfli symbol {3,3,oo}. A rooted polyomino has one external face identified, and chiral pairs are counted as two. a(n) = T(n) in the second Beineke and Pippert link. - _Robert A. Russell_, Mar 20 2024
%D A001764 Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 23.
%D A001764 I. M. H. Etherington, On non-associative combinations, Proc. Royal Soc. Edinburgh, 59 (Part 2, 1938-39), 153-162.
%D A001764 I. M. H. Etherington, Some problems of non-associative combinations (I), Edinburgh Math. Notes, 32 (1940), pp. i-vi. Part II is by A. Erdelyi and I. M. H. Etherington, and is on pages vii-xiv of the same issue.
%D A001764 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, pp. 200, 347. See also the Pólya-Szegő reference.
%D A001764 W. Kuich, Languages and the enumeration of planted plane trees. Nederl. Akad. Wetensch. Proc. Ser. A 73 = Indag. Math. 32, (1970), 268-280.
%D A001764 T. V. Narayana, Lattice Path Combinatorics with Statistical Applications. Univ. Toronto Press, 1979, p. 98.
%D A001764 G. Pólya and G. Szegő, Problems and Theorems in Analysis, Springer-Verlag, New York, Heidelberg, Berlin, 2 vols., 1972, Vol. 1, problem 211, p. 146 with solution on p. 348.
%D A001764 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001764 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001764 G. C. Greubel, <a href="/A001764/b001764.txt">Table of n, a(n) for n = 0..1000</a> [Terms 0 to 100 computed by T. D. Noe; Terms 101 to 1000 by G. C. Greubel, Jan 13 2017]
%H A001764 V. E. Adler and A. B. Shabat, <a href="https://arxiv.org/abs/1810.13198">Volterra chain and Catalan numbers</a>, arXiv:1810.13198 [nlin.SI], 2018.
%H A001764 Ayomikun Adeniran and Lara Pudwell, <a href="https://doi.org/10.54550/ECA2023V3S3R17">Pattern avoidance in parking functions</a>, Enumer. Comb. Appl. 3:3 (2023), Article S2R17.
%H A001764 A. Aggarwal, <a href="http://arxiv.org/abs/1407.5134">Armstrong's Conjecture for (k, mk+1)-Core Partitions</a>, arXiv:1407.5134 [math.CO], 2014.
%H A001764 O. Aichholzer and H. Krasser, <a href="http://www.ist.tugraz.at/files/publications/geometry/ak-psotd-01.ps.gz">The point set order type data base: a collection of applications and results</a>, pp. 17-20 in Abstracts 13th Canadian Conference on Computational Geometry (CCCG '01), Waterloo, Aug. 13-15, 2001.
%H A001764 M. H. Albert, R. E. L. Allred, M. D. Atkinson, H. P. van Ditmarsch, C. C. Handley, and D. A. Holton, <a href="https://doi.org/10.1016/j.disc.2004.08.003">Restricted permutations and queue jumping</a>, Discrete Math. 287 (2004), 129-133.
%H A001764 N. Alexeev and A. Tikhomirov, <a href="http://arxiv.org/abs/1501.04615">Singular Values Distribution of Squares of Elliptic Random Matrices and type-B Narayana Polynomials</a>, arXiv:1501.04615 [math.PR], 2015.
%H A001764 N. V. Alexeev, <a href="https://elibrary.ru/item.asp?id=41167050">Number of trees in a random graph</a>, Probabilistic methods in discrete mathematics, Extended abstracts of the 10th International Petrozavodsk Conference (Russia, 2019), 12-13. (in Russian)
%H A001764 M. Almeida, N. Moreira, and R. Reis, <a href="http://dx.doi.org/10.1016/j.tcs.2007.07.029">Enumeration and generation with a string automata representation</a>, Theor. Comp. Sci. 387 (2007), 93-102, Theorem 6.
%H A001764 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, pp. 337-338.
%H A001764 Joerg Arndt, <a href="http://arxiv.org/abs/1405.6503">Subset-lex: did we miss an order?</a>, arXiv:1405.6503 [math.CO], 2014.
%H A001764 A. Asinowski, B. Hackl, and S. Selkirk, <a href="https://arxiv.org/abs/2007.15562">Down step statistics in generalized Dyck paths</a>, arXiv:2007.15562 [math.CO], 2020.
%H A001764 Jean-Christophe Aval, <a href="http://arxiv.org/abs/0711.0906">Multivariate Fuss-Catalan numbers</a>, arXiv:0711.0906 [math.CO], 2007.
%H A001764 Jean-Christophe Aval, <a href="https://doi.org/10.1016/j.disc.2007.08.100">Multivariate Fuss-Catalan numbers</a>, Discrete Math., 308 (2008), 4660-4669.
%H A001764 I. Bajunaid et al., <a href="http://www.jstor.org/stable/30037599">Function series, Catalan numbers and random walks on trees</a>, Amer. Math. Monthly 112 (2005), 765-785.
%H A001764 P. Balduf, <a href="http://www2.mathematik.hu-berlin.de/~kreimer/wp-content/uploads/PaulMaster">The propagator and diffeomorphisms of an interacting field theory</a>, Master's thesis, submitted to the Institut für Physik, Mathematisch-Naturwissenschaftliche Fakultät, Humboldt-Universtität, Berlin, 2018.
%H A001764 Christian Ballot, <a href="https://www.emis.de/journals/JIS/VOL21/Ballot/ballot30.html">Lucasnomial Fuss-Catalan Numbers and Related Divisibility Questions</a>, J. Int. Seq., 21 (2018), Article 18.6.5.
%H A001764 C. Banderier, M. Bousquet-Mélou, A. Denise, P. Flajolet, D. Gardy and D. Gouyou-Beauchamps, <a href="http://dx.doi.org/10.1016/S0012-365X(01)00250-3">Generating functions for generating trees</a>, Discrete Mathematics 246(1-3) (2002), 29-55.
%H A001764 C. Banderier and D. Merlini, <a href="http://algo.inria.fr/banderier/Papers/infjumps.ps">Lattice paths with an infinite set of jumps</a>, FPSAC02, Melbourne, 2002.
%H A001764 T. Banica, S. Belinschi, M. Capitaine, and B. Collins, <a href="https://arxiv.org/abs/0710.5931">Free Bessel laws</a>, arXiv:0710.5931 [math.PR], 2008.
%H A001764 Paul Barry, <a href="http://dx.doi.org/10.1016/j.laa.2015.10.032">Riordan arrays, generalized Narayana triangles, and series reversion</a>, Linear Algebra and its Applications, 491 (2016), 343-385.
%H A001764 Paul Barry, <a href="https://arxiv.org/abs/1912.11845">Chebyshev moments and Riordan involutions</a>, arXiv:1912.11845 [math.CO], 2019.
%H A001764 Paul Barry, <a href="https://arxiv.org/abs/2001.08799">Characterizations of the Borel triangle and Borel polynomials</a>, arXiv:2001.08799 [math.CO], 2020.
%H A001764 Paul Barry, <a href="https://arxiv.org/abs/2504.09719">Notes on Riordan arrays and lattice paths</a>, arXiv:2504.09719 [math.CO], 2025. See pp. 27, 29.
%H A001764 Y. Baryshnikov, <a href="http://www.math.uiuc.edu/~ymb/texts/stokes.pdf">On Stokes sets</a>, New developments in singularity theory (Cambridge, 2000): 65-86. Kluwer Acad. Publ., Dordrecht, 2001.
%H A001764 L. W. Beineke and R. E. Pippert, Enumerating labeled k-dimensional trees and ball dissections, pp. 12-26 of Proceedings of Second Chapel Hill Conference on Combinatorial Mathematics and Its Applications, University of North Carolina, Chapel Hill, 1970. Reprinted in <a href="http://dx.doi.org/10.1007/BF02330563">Math. Annalen, 191 (1971), 87-98</a>.
%H A001764 L. W. Beineke and R. E. Pippert, <a href="http://dx.doi.org/10.4153/CJM-1974-006-x">Enumerating dissectable polyhedra by their automorphism groups</a>, Canad. J. Math., 26 (1974), 50-67.
%H A001764 Francois Bergeron, <a href="http://arxiv.org/abs/1202.6269">Combinatorics of r-Dyck paths, r-Parking functions, and the r-Tamari lattices</a>, arXiv:1202.6269 [math.CO], 2012.
%H A001764 Olivier Bernardi and Nicolas Bonichon, <a href="http://dx.doi.org/10.1016/j.jcta.2008.05.005">Intervals in Catalan lattices and realizers of triangulations</a>, Journal of Combinatorial Theory, Series A 116:1 (2009), pp. 55-75. See also Bernardi's slides, <a href="http://people.brandeis.edu/~bernardi/slides/slides-Tamari.pdf">Catalan lattices and realizers of triangulations</a> (April 2007).
%H A001764 D. Bevan, D. Levin, P. Nugent, J. Pantone, and L. Pudwell, <a href="http://arxiv.org/abs/1510.08036">Pattern avoidance in forests of binary shrubs</a>, arXiv:1510:08036 [math.CO], 2015.
%H A001764 Allan Bickle, <a href="https://doi.org/10.20429/tag.2024.000105">A Survey of Maximal k-degenerate Graphs and k-Trees</a>, Theory and Applications of Graphs 0 1 (2024) Article 5.
%H A001764 D. Birmajer, J. B. Gil and M. D. Weiner, <a href="http://arxiv.org/abs/1503.05242">Colored partitions of a convex polygon by noncrossing diagonals</a>, arXiv:1503.05242 [math.CO], 2015.
%H A001764 Michel Bousquet and Cédric Lamathe, <a href="https://doi.org/10.46298/dmtcs.420">On symmetric structures of order two</a>, Discrete Math. Theor. Comput. Sci. 10 (2008), 153-176.
%H A001764 M. Bousquet-Mélou and M. Petkovšek, <a href="https://arxiv.org/abs/math/0211432">Walks confined in a quadrant are not always D-finite</a>, arXiv:math/0211432 [math.CO], 2002.
%H A001764 Włodzimierz Bryc, Raouf Fakhfakh, and Wojciech Młotkowski, <a href="https://arxiv.org/abs/1708.05343">Cauchy-Stieltjes families with polynomial variance functions and generalized orthogonality</a>, arXiv:1708.05343 [math.PR], 2017-2019. Also in <a href="http://www.math.uni.wroc.pl/~pms/publicationsArticleRef.php?nr=39.2&nrA=1&ppB=%20237&ppE=%20258">Probability and Mathematical Statistics</a> 39(2) (2019), 237-258.
%H A001764 David Callan and Toufik Mansour, <a href="https://doi.org/10.37236/12720">Ascent sequences avoiding a triple of length-3 patterns</a>, The Electronic Journal of Combinatorics 32(1) (2025), #P1.40. See p. 16.
%H A001764 N. T. Cameron, <a href="https://www.math.hmc.edu/~cameron/dissertation.pdf">Random walks, trees and extensions of Riordan group techniques</a>, Dissertation, Howard University, 2002.
%H A001764 Naiomi Cameron and J. E. McLeod, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/McLeod/mcleod3.html">Returns and Hills on Generalized Dyck Paths</a>, Journal of Integer Sequences, Vol. 19, 2016, #16.6.1.
%H A001764 Peter J. Cameron and Liam Stott, <a href="https://arxiv.org/abs/2010.14902">Trees and cycles</a>, arXiv:2010.14902 [math.CO], 2020. See p. 33.
%H A001764 L. Carlitz, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/11-2/carlitz.pdf">Enumeration of two-line arrays</a>, Fib. Quart., 11(2) (1973), 113-130.
%H A001764 F. Cazals, <a href="http://algo.inria.fr/libraries/autocomb/NCC-html/NCC.html">Combinatorics of Non-Crossing Configurations</a>, Studies in Automatic Combinatorics, Volume II (1997).
%H A001764 Matteo Cervetti and Luca Ferrari, <a href="https://arxiv.org/abs/2009.01024">Pattern avoidance in the matching pattern poset</a>, arXiv:2009.01024 [math.CO], 2020.
%H A001764 W. Y. C. Chen, T. Mansour and S. H. F. Yan, <a href="https://arxiv.org/abs/math/0504342">Matchings avoiding partial patterns</a>, arXiv:math/0504342 [math.CO], 2005.
%H A001764 Malin Christensson, <a href="http://malinc.se/m/ImageTiling.php">Make hyperbolic tilings of images</a>, web page, 2019.
%H A001764 J. Cigler, <a href="http://arxiv.org/abs/1312.2767">Some remarks about q-Chebyshev polynomials and q-Catalan numbers and related results</a>, arXiv:1312.2767 [math.CO], 2013.
%H A001764 B. Collins, I. Nechita, and K. Zyczkowski, <a href="https://arxiv.org/abs/1003.3075">Random graph states, maximal flow and Fuss-Catalan distributions</a>, arXiv:1003.3075 [quant-ph], 2010.
%H A001764 T. C. Copeland, <a href="http://tcjpn.wordpress.com/2014/09/17/compositional-inverse-pairs-the-inviscid-burgers-hopf-equation-and-the-stasheff-associahedra/">Compositional inverse pairs, the Burgers-Hopf equation, and the Stasheff associahedra</a>, 2014.
%H A001764 T. C. Copeland, <a href="http://tcjpn.wordpress.com/2012/06/13/depressed-equations-and-generalized-catalan-numbers/">Discriminating Deltas, Depressed Equations, and Generalized Catalan Numbers</a>, 2012.
%H A001764 S. J. Cyvin, Jianji Wang, J. Brunvoll, Shiming Cao, Ying Li, B. N. Cyvin, and Yugang Wang, <a href="https://doi.org/10.1016/S0022-2860(97)00025-2">Staggered conformers of alkanes: complete solution of the enumeration problem</a>, J. Molec. Struct. 413-414 (1997), 227-239.
%H A001764 S. J. Cyvin et al., <a href="http://dx.doi.org/10.1016/S0022-2860(97)00419-5">Enumeration of staggered conformers of alkanes and monocyclic cycloalkanes</a>, J. Molec. Struct., 445 (1998), 127-13.
%H A001764 Colin Defant, <a href="https://arxiv.org/abs/1904.02627">Catalan Intervals and Uniquely Sorted Permutations</a>, arXiv:1904.02627 [math.CO], 2019.
%H A001764 Colin Defant and N. Kravitz, <a href="https://arxiv.org/abs/1809.09158">Stack-sorting for words</a>, arXiv:1809.09158 [math.CO], 2018.
%H A001764 E. Deutsch, S. Feretic and M. Noy, <a href="http://dx.doi.org/10.1016/S0012-365X(02)00340-0">Diagonally convex directed polyominoes and even trees: a bijection and related issues</a>, Discrete Math., 256 (2002), 645-654.
%H A001764 S. Dulucq, <a href="/A005819/a005819.pdf">Etude combinatoire de problèmes d'énumération, d'algorithmique sur les arbres et de codage par des mots</a>, a thesis presented to l'Université de Bordeaux I, 1987. (Annotated scanned copy)
%H A001764 Jins de Jong, Alexander Hock, and Raimar Wulkenhaar, <a href="https://arxiv.org/abs/1904.11231">Catalan tables and a recursion relation in noncommutative quantum field theory</a>, arXiv:1904.11231 [math-ph], 2019.
%H A001764 Emeric Deutsch and M. Noy, <a href="http://dx.doi.org/10.1016/S0012-365X(01)00366-1">Statistics on non-crossing trees</a>, Discrete Math., 254 (2002), 75-87.
%H A001764 R. Dickau, <a href="http://robertdickau.com/fusscatalan.html">Fuss-Catalan Numbers</a>. Figures of various interpretations.
%H A001764 C. Domb and A. J. Barrett, <a href="http://dx.doi.org/10.1016/0012-365X(74)90081-8">Enumeration of ladder graphs</a>, Discrete Math. 9 (1974), 341-358.
%H A001764 C. Domb and A. J. Barrett, <a href="/A003408/a003408.pdf">Enumeration of ladder graphs</a>, Discrete Math. 9 (1974), 341-358. (Annotated scanned copy)
%H A001764 C. Domb and A. J. Barrett, <a href="/A001764/a001764.pdf">Notes on Table 2 in "Enumeration of ladder graphs"</a>, Discrete Math. 9 (1974), 55. (Annotated scanned copy)
%H A001764 J. A. Eidswick, <a href="http://dx.doi.org/10.1016/0012-365X(89)90266-5">Short factorizations of permutations into transpositions</a>, Disc. Math. 73 (1989) 239-243
%H A001764 Bryan Ek, <a href="https://arxiv.org/abs/1803.10920">Lattice Walk Enumeration</a>, arXiv:1803.10920 [math.CO], 2018.
%H A001764 Bryan Ek, <a href="https://arxiv.org/abs/1804.05933">Unimodal Polynomials and Lattice Walk Enumeration with Experimental Mathematics</a>, arXiv:1804.05933 [math.CO], 2018.
%H A001764 I. M. H. Etherington, <a href="http://www.jstor.org/stable/3605743">Non-associate powers and a functional equation</a>, Math. Gaz. 21 (1937), 36-39; addendum 21 (1937), 153.
%H A001764 I. M. H. Etherington, <a href="http://dx.doi.org/10.1017/S0950184300002639">Some problems of non-associative combinations</a>, Edinburgh Math. Notes, 32 (1940), 1-6.
%H A001764 I. M. H. Etherington, <a href="/A000108/a000108_14.pdf">Some problems of non-associative combinations (I)</a>, Edinburgh Math. Notes, 32 (1940), pp. i-vi. [Annotated scanned copy]. Part II [not scanned] is by A. Erdelyi and I. M. H. Etherington, and it is on pages vii-xiv of the same issue.
%H A001764 Jishe Feng, <a href="https://arxiv.org/abs/1810.09170">The Hessenberg matrices and Catalan and its generalized numbers</a>, arXiv:1810.09170 [math.CO], 2018. See p. 4.
%H A001764 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; see page 486.
%H A001764 N. Gabriel, K. Peske, L. Pudwell, and S. Tay, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Pudwell/pudwell.html">Pattern Avoidance in Ternary Trees</a>, J. Int. Seq. 15 (2012), #12.1.5
%H A001764 I. Gessel and G. Xin, <a href="https://arxiv.org/abs/math/0505217">The generating function of ternary trees and continued fractions</a>, arXiv:math/0505217 [math.CO], 2005.
%H A001764 S. Goldstein, J. L. Lebowitz, E. R. Speer, <a href="https://arxiv.org/abs/2003.04995">The Discrete-Time Facilitated Totally Asymmetric Simple Exclusion Process</a>, arXiv:2003.04995 [math-ph], 2020.
%H A001764 N. S. S. Gu, N. Y. Li and T. Mansour, <a href="http://dx.doi.org/10.1016/j.disc.2007.04.007">2-Binary trees: bijections and related issues</a>, Discr. Math., 308 (2008), 1209-1221.
%H A001764 Nancy S.S. Gu and Helmut Prodinger, <a href="https://arxiv.org/abs/2007.02142">A bijection between two subfamilies of Motzkin paths</a>, arXiv:2007.02142 [math.CO], 2020.
%H A001764 F. Harary, E. M. Palmer, R. C. Read, <a href="/A000108/a000108_20.pdf">On the cell-growth problem for arbitrary polygons, computer printout, circa 1974</a>
%H A001764 Tian-Xiao He, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL25/He/he49.html">The Vertical Recursive Relation of Riordan Arrays and Its Matrix Representation</a>, J. Int. Seq., Vol. 25 (2022), Article 22.9.5.
%H A001764 T.-X. He, L. W. Shapiro, <a href="http://dx.doi.org/10.1016/j.laa.2017.06.025">Fuss-Catalan matrices, their weighted sums, and stabilizer subgroups of the Riordan group</a>, Lin. Alg. Applic. 532 (2017) 25-41, Fuss-Catalan Number (F_3)_n
%H A001764 V. E. Hoggatt, Jr., <a href="/A001628/a001628.pdf">Letters to N. J. A. Sloane, 1974-1975</a>.
%H A001764 V. E. Hoggatt, Jr., <a href="/A005676/a005676.pdf">7-page typed letter to N. J. A. Sloane with suggestions for new sequences</a>, circa 1977.
%H A001764 V. E. Hoggatt, Jr. and M. Bicknell, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/14-5/hoggatt1.pdf">Catalan and related sequences arising from inverses of Pascal's triangle matrices</a>, Fib. Quart., 14 (1976), 395-405.
%H A001764 Vera M. Hur, M. A. Johnson, and J. L. Martin, <a href="https://arxiv.org/abs/1609.02189">Oscillation estimates of eigenfunctions via the combinatorics of noncrossing partitions</a>, arXiv:1609.02189 [math.SP], 2016.
%H A001764 Hsien-Kuei Hwang, Mihyun Kang, and Guan-Huei Duh, <a href="https://doi.org/10.4230/LIPIcs.AofA.2018.29">Asymptotic Expansions for Sub-Critical Lagrangean Forms</a>, LIPIcs Proceedings of Analysis of Algorithms 2018, Vol. 110. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2018.
%H A001764 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=53">Encyclopedia of Combinatorial Structures 53</a>.
%H A001764 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=285">Encyclopedia of Combinatorial Structures 285</a>.
%H A001764 C. Ji and J. Propp, <a href="https://arxiv.org/abs/1805.03608">Brussels Sprouts, Noncrossing Trees, and Parking Functions</a>, arXiv:1805.03608 [math.CO], 2018.
%H A001764 J. Jong, A. Hock, and R. Wulkenhaar <a href="https://arxiv.org/abs/1904.11231">Catalan tables and a recursion relation in noncommutative quantum field theory</a>, arXiv:1904.11231 [math-ph], 2019.
%H A001764 A. V. Kitaev, <a href="https://arxiv.org/abs/1809.00122">Meromorphic Solution of the Degenerate Third Painlevé Equation Vanishing at the Origin</a>, arXiv:1809.00122 [math.CA], 2018.
%H A001764 S. Kitaev and A. de Mier, <a href="http://arxiv.org/abs/1210.2618">Enumeration of fixed points of an involution on beta(1, 0)-trees</a>, arXiv:1210.2618 [math.CO], 2012.
%H A001764 Don Knuth, <a href="https://www.youtube.com/watch?v=P4AaGQIo0HY">20th Anniversary Christmas Tree Lecture</a>.
%H A001764 G. Kreweras, <a href="http://dx.doi.org/10.1016/0012-365X(72)90041-6">Sur les partitions non croisées d'un cycle</a>, (French) Discrete Math. 1(4) (1972), 333-350. MR0309747 (46 #8852).
%H A001764 V. N. Krishnachandran, <a href="https://arxiv.org/abs/2411.08296">On the computation of the arcsin function in the Kerala school of astronomy and mathematics</a>, arXiv:2411.08296 [math.HO], 2024. See pp. 3, 13, 25.
%H A001764 D. V. Kruchinin, <a href="http://dx.doi.org/10.1186/s13662-014-0347-9">On solving some functional equations</a>, Advances in Difference Equations (2015), 2015:17.
%H A001764 Dmitry V. Kruchinin and Vladimir V. Kruchinin, <a href="http://www.emis.de/journals/JIS/VOL18/Kruchinin/kruch9.html">A Generating Function for the Diagonal T_{2n,n} in Triangles</a>, Journal of Integer Sequences, 18 (2015), Article 15.4.6.
%H A001764 Markus Kuba and Alois Panholzer, <a href="http://dx.doi.org/10.1016/j.disc.2012.07.011">Enumeration formulas for pattern restricted Stirling permutations</a>, Discrete Math. 312(21) (2012), 3179--3194. MR2957938. - From _N. J. A. Sloane_, Sep 25 2012
%H A001764 Wolfdieter Lang, <a href="/A001764/a001764fig.pdf">Ternary trees with n = 1, 2, 3 and 4 vertices</a>.
%H A001764 Ho-Hon Leung and Thotsaporn "Aek" Thanatipanonda, <a href="https://arxiv.org/abs/1903.03274">A Probabilistic Two-Pile Game</a>, arXiv:1903.03274 [math.CO], 2019.
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%H A001764 Lun Lv and Sabrina X.M. Pang, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v18i1p107">Reduced Decompositions of Matchings</a>, Electronic Journal of Combinatorics 18 (2011), #P107.
%H A001764 D. Merlini, R. Sprugnoli and M. C. Verri, <a href="http://dx.doi.org/10.1006/jcta.2002.3273">The tennis ball problem</a>, J. Combin. Theory, A 99 (2002), 307-344, (T_n for s=3).
%H A001764 Hugo Mlodecki, <a href="https://arxiv.org/abs/2205.13949">Decompositions of packed words and self duality of Word Quasisymmetric Functions</a>, arXiv:2205.13949 [math.CO], 2022. See Table 4 p. 20.
%H A001764 W. Mlotkowski, M. Nowak, K. Penson, and K. Zyczkowski, <a href="https://arxiv.org/abs/1407.1282">Spectral density of generalized Wishart matrices and free multiplicative convolution</a>, arXiv preprint arXiv:1407.1282 [math-ph], 2015.
%H A001764 W. Mlotkowski and K. A. Penson, <a href="http://arxiv.org/abs/1304.6544">The probability measure corresponding to 2-plane trees</a>, arXiv:1304.6544 [math.PR], 2013.
%H A001764 Hanna Mularczyk, <a href="https://arxiv.org/abs/1908.04025">Lattice Paths and Pattern-Avoiding Uniquely Sorted Permutations</a>, arXiv:1908.04025 [math.CO], 2019.
%H A001764 Emanuele Munarini, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Munarini/muna4.html">Shifting Property for Riordan, Sheffer and Connection Constants Matrices</a>, Journal of Integer Sequences, 20 (2017), Article 17.8.2.
%H A001764 H. Niederhausen, <a href="https://doi.org/10.37236/1649">Catalan Traffic at the Beach</a>, Electronic Journal of Combinatorics, 9 (2002), #R33.
%H A001764 J.-C. Novelli, J.-Y. Thibon, <a href="http://arxiv.org/abs/1403.5962">Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions</a>, arXiv:1403.5962 [math.CO], 2014.
%H A001764 M. Noy, <a href="http://dx.doi.org/10.1016/S0012-365X(97)00121-0">Enumeration of noncrossing trees on a circle</a>, Discrete Math., 180 (1998), 301-313.
%H A001764 R. N. Onody and U. P. C. Neves, <a href="https://doi.org/10.1088/0305-4470/25/24/014">Series Expansion of the Directed Percolation Probability</a>, J. Phys. A 25 (1992), 6609-6615.
%H A001764 A. Panholzer and H. Prodinger, <a href="http://doi.org/10.1016/S0012-365X(01)00282-5">Bijections for ternary trees and non-crossing trees</a>, Discrete Math., 250 (2002), 181-195.
%H A001764 K. A. Penson and A. I. Solomon, <a href="http://arXiv.org/abs/quant-ph/0111151">Coherent states from combinatorial sequences</a>, arXiv:quant-ph/0111151, 2001.
%H A001764 K. H. Pilehrood and T. H. Pilehrood, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Pilehrood/pile5.html">Jacobi Polynomials and Congruences Involving Some Higher-Order Catalan Numbers and Binomial Coefficients</a>, J. Int. Seq. 18 (2015), #15.11.7.
%H A001764 Helmut Prodinger, <a href="https://doi.org/10.1016/j.disc.2008.01.035">A simple bijection between a subclass of 2-binary trees and ternary trees</a>, Discrete Mathematics 309(4) (2009), 959-961.
%H A001764 Helmut Prodinger, <a href="https://arxiv.org/abs/2004.04215">Generating functions for a lattice path model introduced by Deutsch</a>, arXiv:2004.04215 [math.CO], 2020.
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%H A001764 Jocelyn Quaintance, <a href="http://dx.doi.org/10.1016/j.disc.2006.09.041">Combinatoric Enumeration of Two-Dimensional Proper Arrays</a>, Discrete Math., 307 (2007), 1844-1864.
%H A001764 J. Riordan, <a href="/A001850/a001850_2.pdf">Letter, Jul 06 1978</a>.
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%H A001764 A. Schuetz and G. Whieldon, <a href="http://arxiv.org/abs/1401.7194">Polygonal Dissections and Reversions of Series</a>, arXiv:1401.7194 [math.CO], 2014.
%H A001764 Makoto Sekiyama, Toshiya Ohtsuki, and Hiroshi Yamamoto, <a href="https://doi.org/10.7566/JPSJ.86.104003">Analytical Solution of Smoluchowski Equations in Aggregation-Fragmentation Processes</a>, Journal of the Physical Society of Japan, 86.10, id 104003 (2017).
%H A001764 Michael Somos, <a href="http://cis.csuohio.edu/~somos/nwic.html">Number Walls in Combinatorics</a>.
%H A001764 B. Sury, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Sury/sury31.html">Generalized Catalan numbers: linear recursion and divisibility</a>, JIS 12 (2009), #09.7.5
%H A001764 L. Takacs, <a href="http://www.appliedprobability.org/data/files/TMS%20articles/18_1_1.pdf">Enumeration of rooted trees and forests</a>, Math. Scientist 18 (1993), 1-10, esp. Eq. (5).
%H A001764 N. J. Wildberger and Dean Rubine, <a href="https://doi.org/10.1080/00029890.2025.2460966">A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode</a>, Amer. Math. Monthly (2025). See sections 11.3 and 12.
%H A001764 Aaron Williams, <a href="https://www.researchgate.net/publication/373140133_Pattern_Avoidance_for_k-Catalan_Sequences">Pattern Avoidance for k-Catalan Sequences</a>, Proc. 21st Int'l Conf. Permutation Patterns (2023).
%H A001764 S. Yakoubov, <a href="http://arxiv.org/abs/1310.2979">Pattern Avoidance in Extensions of Comb-Like Posets</a>, arXiv:1310.2979 [math.CO], 2013.
%H A001764 Sheng-Liang Yang, L.-J. Wang, <a href="https://ajc.maths.uq.edu.au/pdf/64/ajc_v64_p420.pdf">Taylor expansions for the m-Catalan numbers</a>, Australasian Journal of Combinatorics, 64(3) (2016), 420-431.
%H A001764 Anssi Yli-Jyra, <a href="http://dx.doi.org/10.1007/978-3-642-30773-7_10">On Dependency Analysis via Contractions and Weighted FSTs</a>, in Shall We Play the Festschrift Game?, Springer, 2012, pp. 133-158.
%H A001764 S.-n. Zheng and S.-l. Yang, <a href="http://dx.doi.org/10.1155/2014/848374">On the-Shifted Central Coefficients of Riordan Matrices</a>, Journal of Applied Mathematics, Volume 2014, Article ID 848374, 8 pages.
%H A001764 Jian Zhou, <a href="https://arxiv.org/abs/1810.03883">Fat and Thin Emergent Geometries of Hermitian One-Matrix Models</a>, arXiv:1810.03883 [math-ph], 2018.
%H A001764 <a href="/index/Cor#core">Index entries for "core" sequences</a>
%H A001764 <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F A001764 From _Karol A. Penson_, Nov 08 2001: (Start)
%F A001764 G.f.: (2/sqrt(3*x))*sin((1/3)*arcsin(sqrt(27*x/4))).
%F A001764 E.g.f.: hypergeom([1/3, 2/3], [1, 3/2], 27/4*x).
%F A001764 Integral representation as n-th moment of a positive function on [0, 27/4]: a(n) = Integral_{x=0..27/4} (x^n*((1/12) * 3^(1/2) * 2^(1/3) * (2^(1/3)*(27 + 3 * sqrt(81 - 12*x))^(2/3) - 6 * x^(1/3))/(Pi * x^(2/3)*(27 + 3 * sqrt(81 - 12*x))^(1/3)))), n >= 0. This representation is unique. (End)
%F A001764 G.f. A(x) satisfies A(x) = 1+x*A(x)^3 = 1/(1-x*A(x)^2) [Cyvin (1998)]. - _Ralf Stephan_, Jun 30 2003
%F A001764 a(n) = n-th coefficient in expansion of power series P(n), where P(0) = 1, P(k+1) = 1/(1 - x*P(k)^2).
%F A001764 G.f. Rev(x/c(x))/x, where c(x) is the g.f. of A000108 (Rev=reversion of). - _Paul Barry_, Mar 26 2010
%F A001764 From _Gary W. Adamson_, Jul 07 2011: (Start)
%F A001764 Let M = the production matrix:
%F A001764 1, 1
%F A001764 2, 2, 1
%F A001764 3, 3, 2, 1
%F A001764 4, 4, 3, 2, 1
%F A001764 5, 5, 4, 3, 2, 1
%F A001764 ...
%F A001764 a(n) = upper left term in M^n. Top row terms of M^n = (n+1)-th row of triangle A143603, with top row sums generating A006013: (1, 2, 7, 30, 143, 728, ...). (End)
%F A001764 Recurrence: a(0)=1; a(n) = Sum_{i=0..n-1, j=0..n-1-i} a(i)a(j)a(n-1-i-j) for n >= 1 (counts ternary trees by subtrees of the root). - _David Callan_, Nov 21 2011
%F A001764 G.f.: 1 + 6*x/(Q(0) - 6*x); Q(k) = 3*x*(3*k + 1)*(3*k + 2) + 2*(2*(k^2) + 5*k +3) - 6*x*(2*(k^2) + 5*k + 3)*(3*k + 4)*(3*k + 5)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Nov 27 2011
%F A001764 D-finite with recurrence: 2*n*(2n+1)*a(n) - 3*(3n-1)*(3n-2)*a(n-1) = 0. - _R. J. Mathar_, Dec 14 2011
%F A001764 REVERT transform of A115140. BINOMIAL transform is A188687. SUMADJ transform of A188678. HANKEL transform is A051255. INVERT transform of A023053. INVERT transform is A098746. - _Michael Somos_, Apr 07 2012
%F A001764 (n + 1) * a(n) = A174687(n).
%F A001764 G.f.: F([2/3,4/3], [3/2], 27/4*x) / F([2/3,1/3], [1/2], (27/4)*x) where F() is the hypergeometric function. - _Joerg Arndt_, Sep 01 2012
%F A001764 a(n) = binomial(3*n+1, n)/(3*n+1) = A062993(n+1,1). - _Robert FERREOL_, Apr 03 2015
%F A001764 a(n) = A258708(2*n,n) for n > 0. - _Reinhard Zumkeller_, Jun 23 2015
%F A001764 0 = a(n)*(-3188646*a(n+2) + 20312856*a(n+3) - 11379609*a(n+4) + 1437501*a(n+5)) + a(n+1)*(177147*a(n+2) - 2247831*a(n+3) + 1638648*a(n+4) - 238604*a(n+5)) + a(n+2)*(243*a(n+2) + 31497*a(n+3) - 43732*a(n+4) + 8288*a(n+5)) for all integer n. - _Michael Somos_, Jun 03 2016
%F A001764 a(n) ~ 3^(3*n + 1/2)/(sqrt(Pi)*4^(n+1)*n^(3/2)). - _Ilya Gutkovskiy_, Nov 21 2016
%F A001764 Given g.f. A(x), then A(1/8) = -1 + sqrt(5), A(2/27) = (-1 + sqrt(3))*3/2, A(4/27) = 3/2, A(3/64) = -2 + 2*sqrt(7/3), A(5/64) = (-1 + sqrt(5))*2/sqrt(5), etc. A(n^2/(n+1)^3) = (n+1)/n if n > 1. - _Michael Somos_, Jul 17 2018
%F A001764 From _Peter Bala_, Sep 14 2021: (Start)
%F A001764 A(x) = exp( Sum_{n >= 1} (1/3)*binomial(3*n,n)*x^n/n ).
%F A001764 The sequence defined by b(n) := [x^n] A(x)^n = A224274(n) for n >= 1 and satisfies the congruence b(p) == b(1) (mod p^3) for prime p >= 3. Cf. A060941. (End)
%F A001764 G.f.: 1/sqrt(B(x)+(1-6*x)/(9*B(x))+1/3), with B(x):=((27*x^2-18*x+2)/54-(x*sqrt((-(4-27*x))*x))/(2*3^(3/2)))^(1/3). - _Vladimir Kruchinin_, Sep 28 2021
%F A001764 x*A'(x)/A(x) = (A(x) - 1)/(- 2*A(x) + 3) = x + 5*x^2 + 28*x^3 + 165*x^4 + ... is the o.g.f. of A025174. Cf. A002293 - A002296. - _Peter Bala_, Feb 04 2022
%F A001764 a(n) = hypergeom([1 - n, -2*n], [2], 1). Row sums of A108767. - _Peter Bala_, Aug 30 2023
%F A001764 G.f.: z*exp(3*z*hypergeom([1, 1, 4/3, 5/3], [3/2, 2, 2], (27*z)/4)) + 1.
%F A001764 - _Karol A. Penson_, Dec 19 2023
%F A001764 G.f.: hypergeometric([1/3, 2/3], [3/2], (3^3/2^2)*x). See the e.g.f. above. - _Wolfdieter Lang_, Feb 04 2024
%F A001764 a(n) = (3*n)! / (n!*(2*n+1)!). - _Allan Bickle_, Feb 20 2024
%F A001764 Sum_{n >= 0} a(n)*x^n/(1 + x)^(3*n+1) = 1. See A316371 and A346627. - _Peter Bala_, Jun 02 2024
%F A001764 G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^5). - _Seiichi Manyama_, Jun 16 2025
%e A001764 a(2) = 3 because the only dissections with 5 edges are given by a square dissected by any of the two diagonals and the pentagon with no dissecting diagonal.
%e A001764 G.f. = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 + 7752*x^7 + 43263*x^8 + ...
%p A001764 A001764 := n->binomial(3*n,n)/(2*n+1): seq(A001764(n), n=0..25);
%p A001764 with(combstruct): BB:=[T,{T=Prod(Z,F),F=Sequence(B),B=Prod(F,Z,F)}, unlabeled]:seq(count(BB,size=i),i=0..22); # _Zerinvary Lajos_, Apr 22 2007
%p A001764 with(combstruct):BB:=[S, {B = Prod(S,S,Z), S = Sequence(B)}, labelled]: seq(count(BB, size=n)/n!, n=0..21); # _Zerinvary Lajos_, Apr 25 2008
%p A001764 n:=30:G:=series(RootOf(g = 1+x*g^3, g),x=0,n+1):seq(coeff(G,x,k),k=0..n); # _Robert FERREOL_, Apr 03 2015
%p A001764 alias(PS=ListTools:-PartialSums): A001764List := proc(m) local A, P, n;
%p A001764 A := [1,1]; P := [1]; for n from 1 to m - 2 do P := PS(PS([op(P), P[-1]]));
%p A001764 A := [op(A), P[-1]] od; A end: A001764List(25); # _Peter Luschny_, Mar 26 2022
%t A001764 InverseSeries[Series[y-y^3, {y, 0, 24}], x] (* then a(n)=y(2n+1)=ways to place non-crossing diagonals in convex (2n+4)-gon so as to create only quadrilateral tiles *) (* _Len Smiley_, Apr 08 2000 *)
%t A001764 Table[Binomial[3n,n]/(2n+1),{n,0,25}] (* _Harvey P. Dale_, Jul 24 2011 *)
%o A001764 (PARI) {a(n) = if( n<0, 0, (3*n)! / n! / (2*n + 1)!)};
%o A001764 (PARI) {a(n) = if( n<0, 0, polcoeff( serreverse( x - x^3 + O(x^(2*n + 2))), 2*n + 1))};
%o A001764 (PARI) {a(n) = my(A); if( n<0, 0, A = 1 + O(x); for( m=1, n, A = 1 + x * A^3); polcoeff(A, n))};
%o A001764 (PARI) b=vector(22);b[1]=1;for(n=2,22,for(i=1,n-1,for(j=1,n-1,for(k=1,n-1,if((i-1)+(j-1)+(k-1)-(n-2),NULL,b[n]=b[n]+b[i]*b[j]*b[k])))));a(n)=b[n+1]; print1(a(0));for(n=1,21,print1(", ",a(n))) \\ _Gerald McGarvey_, Oct 08 2008
%o A001764 (PARI) Vec(1 + serreverse(x / (1+x)^3 + O(x^30))) \\ _Gheorghe Coserea_, Aug 05 2015
%o A001764 (Sage)
%o A001764 def A001764_list(n) :
%o A001764 D = [0]*(n+1); D[1] = 1
%o A001764 R = []; b = false; h = 1
%o A001764 for i in range(2*n) :
%o A001764 for k in (1..h) : D[k] += D[k-1]
%o A001764 if not b : R.append(D[h])
%o A001764 else : h += 1
%o A001764 b = not b
%o A001764 return R
%o A001764 A001764_list(22) # _Peter Luschny_, May 03 2012
%o A001764 (Magma) [Binomial(3*n,n)/(2*n+1): n in [0..30]]; // _Vincenzo Librandi_, Sep 04 2014
%o A001764 (Haskell)
%o A001764 a001764 n = a001764_list !! n
%o A001764 a001764_list = 1 : [a258708 (2 * n) n | n <- [1..]]
%o A001764 -- _Reinhard Zumkeller_, Jun 23 2015
%o A001764 (GAP) List([0..25],n->Binomial(3*n,n)/(2*n+1)); # _Muniru A Asiru_, Oct 31 2018
%o A001764 (Python)
%o A001764 from math import comb
%o A001764 def A001764(n): return comb(3*n,n)//(2*n+1) # _Chai Wah Wu_, Nov 10 2022
%Y A001764 Cf. A001762, A001763, A002294 - A002296, A006013, A025174, A063548, A064017, A072247, A072248, A134264, A143603, A258708, A256311, A188687 (binomial transform), A346628 (inverse binomial transform).
%Y A001764 A column of triangle A102537.
%Y A001764 Bisection of A047749 and A047761.
%Y A001764 Row sums of triangles A108410 and A108767.
%Y A001764 Second column of triangle A062993.
%Y A001764 Mod 3 = A113047.
%Y A001764 2D Polyominoes: A005034 (oriented), A005036 (unoriented), A369315 (chiral), A047749 (achiral), A000108 {3,oo}, A002293 {5,oo}.
%Y A001764 3D Polyominoes: A007173 (oriented), A027610 (unoriented), A371350 (chiral), A371351 (achiral).
%Y A001764 Cf. A130564 (for C(k, n) cases).
%K A001764 easy,nonn,nice,core
%O A001764 0,3
%A A001764 _N. J. A. Sloane_