cp's OEIS Frontend

A002293 Number of dissections of a polygon: binomial(4*n, n)/(3*n + 1).

%I A002293 M3587 N1454 #468 Jun 16 2025 08:46:51
%S A002293 1,1,4,22,140,969,7084,53820,420732,3362260,27343888,225568798,
%T A002293 1882933364,15875338990,134993766600,1156393243320,9969937491420,
%U A002293 86445222719724,753310723010608,6594154339031800,57956002331347120,511238042454541545
%N A002293 Number of dissections of a polygon: binomial(4*n, n)/(3*n + 1).
%C A002293 The number of rooted loopless n-edge maps in the plane (planar with a distinguished outside face). - _Valery A. Liskovets_, Mar 17 2005
%C A002293 Number of lattice paths from (1,0) to (3*n+1,n) which, starting from (1,0), only utilize the steps +(1,0) and +(0,1) and additionally, the paths lie completely below the line y = (1/3)*x (i.e., if (a,b) is in the path, then b < a/3). - Joseph Cooper (jecooper(AT)mit.edu), Feb 07 2006
%C A002293 Number of length-n restricted growth strings (RGS) [s(0), s(1), ..., s(n-1)] where s(0) = 0 and s(k) <= s(k-1) + 3, see fxtbook link below. - _Joerg Arndt_, Apr 08 2011
%C A002293 From _Wolfdieter Lang_, Sep 14 2007: (Start)
%C A002293 a(n), n >= 1, enumerates quartic trees (rooted, ordered, incomplete) with n vertices (including the root).
%C A002293 Pfaff-Fuss-Catalan sequence C^{m}_n for m = 4. See the Graham et al. reference, p. 347. eq. 7.66. (Second edition, p. 361, eq. 7.67.) See also the Pólya-Szegő reference.
%C A002293 Also 4-Raney sequence. See the Graham et al. reference, pp. 346-347.
%C A002293 (End)
%C A002293 Bacher: "We describe the statistics of checkerboard triangulations obtained by coloring black every other triangle in triangulations of convex polygons." The current sequence (A002293) occurs on p. 12 as one of two "extremal sequences" of an array of coefficients of polynomials, whose generating functions are given in terms of hypergeometric functions. - _Jonathan Vos Post_, Oct 05 2007
%C A002293 A generating function in terms of a (labyrinthine) solution to a depressed quartic equation is given in the Copeland link for signed A005810. With D(z,t) that g.f., a g.f. for signed A002293 is {[-1+1/D(z,t)]/(4t)}^(1/3). - _Tom Copeland_, Oct 10 2012
%C A002293 For a relation to the inviscid Burgers's equation, see A001764. - _Tom Copeland_, Feb 15 2014
%C A002293 For relations to compositional inversion, the Legendre transform, and convex geometry, see the Copeland, the Schuetz and Whieldon, and the Gross (p. 58) links. - _Tom Copeland_, Feb 21 2017 (See also Gross et al. in A062994. - _Tom Copeland_, Dec 24 2019)
%C A002293 This is the number of A'Campo bicolored forests of degree n and co-dimension 0. This can be shown using generating functions or a combinatorial approach. See Combe and Jugé link below. - _Noemie Combe_, Feb 28 2017
%C A002293 Conjecturally, a(n) is the number of 3-uniform words over the alphabet [n] that avoid the patterns 231 and 221 (see the Defant and Kravitz link). - _Colin Defant_, Sep 26 2018
%C A002293 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. Cf. A001764. - _Tom Copeland_, Dec 13 2019
%C A002293 a(n) is the total number of down steps before the first up step in all 3_1-Dyck paths of length 4*n. A 3_1-Dyck path is a lattice path with steps (1, 3), (1, -1) that starts and ends at y = 0 and stays above the line y = -1. - _Sarah Selkirk_, May 10 2020
%C A002293 a(n) is the number of pairs (A<=B) of noncrossing partitions of [2n] such that every block of A has exactly two elements. In fact, it is proved that a(n) is the number of planar tied arc diagrams with n arcs (see Aicardi link below). A planar diagram with n arcs represents a noncrossing partition A of [2n] with n blocks, each block containing the endpoints of one arc; each tie connects two arcs, so that the ties define a partition B >= A: the endpoints of two arcs connected by a tie belong to the same block of B. Ties do not cross arcs nor other ties iff B has a planar diagram, i.e., B is a noncrossing partition. - _Francesca Aicardi_, Nov 07 2022
%C A002293 Dropping the initial 1 (starting 1, 4, 22 with offset 1) yields the REVERT transformation 1, -4 ,10, -20, 35.. essentially A000292 without leading 0. - _R. J. Mathar_, Aug 17 2023
%C A002293 Number of rooted polyominoes composed of n pentagonal cells of the hyperbolic regular tiling with Schläfli symbol {5,oo}. A rooted polyomino has one external edge identified, and chiral pairs are counted as two. A stereographic projection of the {5,oo} tiling on the Poincaré disk can be obtained via the Christensson link. - _Robert A. Russell_, Jan 27 2024
%C A002293 This is instance k = 4 of the generalized Catalan family {C(k, n)}_{n>=0} given in a comment of A130564. - _Wolfdieter Lang_, Feb 05 2024
%C A002293 a(n) is the cardinality of the planar ramified Jones monoid PR(J_n). - _Diego Arcis_, Nov 21 2024
%D A002293 Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 23.
%D A002293 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, pp. 200, 347.
%D A002293 Peter Hilton and Jean Pedersen, Catalan numbers, their generalization, and their uses, Math. Intelligencer 13 (1991), no. 2, 64-75.
%D A002293 V. A. Liskovets and T. R. Walsh, Enumeration of unrooted maps on the plane, Rapport technique, UQAM, No. 2005-01, Montreal, Canada, 2005.
%D A002293 G. Pólya and G. Szegő, Problems and Theorems in Analysis, Springer-Verlag, Heidelberg, New York, 2 vols., 1972, Vol. 1, problem 211, p. 146 with solution on p. 348.
%D A002293 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002293 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002293 G. C. Greubel, <a href="/A002293/b002293.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 14 2017]
%H A002293 Norbert A'Campo, <a href="https://arxiv.org/abs/1702.05885">Signatures of monic polynomials</a>, arXiv:1702.05885 [math.AG], 2017.
%H A002293 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 A002293 Francesca Aicardi, <a href="https://arxiv.org/abs/2011.14628">Catalan triangle and tied arc diagrams</a>, arXiv:2011.14628 [math.CO], 2020.
%H A002293 Francesca Aicardi, Diego Arcis, and Jesús Juyumaya, <a href="https://www.doi.org/10.17323/1609-4514-2024-24-3-321-355">Ramified inverse and planar monoids</a>, Mosc Math J, 24(3):321-355, 9 2024.
%H A002293 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, Theor. 6
%H A002293 T. Anderson, T. B. McLean, H. Pajoohesh, and C. Smith, <a href="https://doi.org/10.1016/j.ejc.2009.01.005">The combinatorics of all regular flexagons</a>, Eu. J. Combinat. 31 (2010) 72-80, Theorem 2.
%H A002293 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, pp. 337-338
%H A002293 Joerg Arndt, <a href="http://arxiv.org/abs/1405.6503">Subset-lex: did we miss an order?</a>, arXiv:1405.6503 [math.CO], 2014-2015.
%H A002293 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 A002293 Roland Bacher, <a href="http://arXiv.org/abs/0710.0960">Fair Triangulations</a>, arXiv:0710.0960 [math.CO], 2007.
%H A002293 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-Universtät, Berlin, 2018.
%H A002293 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 A002293 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 A002293 Paul Barry, <a href="https://arxiv.org/abs/2104.05593">On the Gap-sum and Gap-product Sequences of Integer Sequences</a>, arXiv:2104.05593 [math.CO], 2021.
%H A002293 Paul Barry, <a href="https://arxiv.org/abs/2505.16718">d-orthogonal polynomials, Fuss-Catalan matrices and lattice paths</a>, arXiv:2505.16718 [math.CO], 2025. See p. 2.
%H A002293 M. Bernstein and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0205301">Some canonical sequences of integers</a>, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210.
%H A002293 M. Bernstein and N. J. A. Sloane, <a href="/A003633/a003633_1.pdf">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
%H A002293 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 preprint arXiv:1510:08036 [math.CO], 2015.
%H A002293 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 A002293 T. Daniel Brennan, Christian Ferko, and Savdeep Sethi, <a href="https://arxiv.org/abs/1912.12389">A Non-Abelian Analogue of DBI from T₸</a>, arXiv:1912.12389 [hep-th], 2019. See also <a href="https://doi.org/10.21468/SciPostPhys.8.4.052">SciPost Phys.</a> Vol. 8 (2020), 052.
%H A002293 Wun-Seng Chou, Tian-Xiao He, and Peter J.-S. Shiue, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/He/he61.html">On the Primality of the Generalized Fuss-Catalan Numbers</a>, J. Int. Seqs., 21 (2018), #18.2.1.
%H A002293 Malin Christensson, <a href="http://malinc.se/m/ImageTiling.php">Make hyperbolic tilings of images</a>, web page, 2019.
%H A002293 J. Cigler, <a href="http://homepage.univie.ac.at/Johann.Cigler/preprints/chebyshev-survey.pdf">Some remarks about q-Chebyshev polynomials and q-Catalan numbers and related results</a>, 2013.
%H A002293 N. Combe and V. Jugé, <a href="https://arxiv.org/abs/1702.07672">Counting bi-colored A'Campo forests</a>, arXiv:1702.07672 [math.AG], 2017.
%H A002293 Tom 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 A002293 Olivier Danvy, <a href="https://arxiv.org/abs/2412.03127">Summa Summarum: Moessner's Theorem without Dynamic Programming</a>, arXiv:2412.03127 [cs.DM], 2024. See p. 25.
%H A002293 C. Defant and N. Kravitz, <a href="https://arxiv.org/abs/1809.09158">Stack-sorting for words</a>, arXiv:1809.09158 [math.CO], 2018.
%H A002293 Isaac DeJager, Madeleine Naquin, and Frank Seidl, <a href="https://www.valpo.edu/mathematics-statistics/files/2019/08/Drube2019.pdf">Colored Motzkin Paths of Higher Order</a>, VERUM 2019.
%H A002293 Bryan Ek, <a href="https://arxiv.org/abs/1803.10920">Lattice Walk Enumeration</a>, arXiv:1803.10920 [math.CO], 2018.
%H A002293 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 A002293 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 A002293 M. Gross, <a href="https://arxiv.org/abs/1212.4220">Mirror symmetry and the Strominger-Yau-Zaslow conjecture</a>, arXiv:1212.4220 [math.AG], p. 58, 2013.
%H A002293 F. Harary, E. M. Palmer, and R. C. Read, <a href="/A000108/a000108_20.pdf">On the cell-growth problem for arbitrary polygons, computer printout, circa 1974</a>.
%H A002293 F. Harary, E. M. Palmer and R. C. Read, <a href="http://dx.doi.org/10.1016/0012-365X(75)90041-2">On the cell-growth problem for arbitrary polygons</a>, Discr. Math. 11 (1975), 371-389.
%H A002293 Clemens Heuberger, Sarah J. Selkirk, and Stephan Wagner, <a href="https://arxiv.org/abs/2204.14023">Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo k</a>, arXiv:2204.14023 [math.CO], 2022.
%H A002293 Forrest M. Hilton, <a href="https://arxiv.org/abs/2408.01353">Finite Dynamical Laminations</a>, arXiv:2408.01353 [math.DS], 2024. See p. 7.
%H A002293 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 A002293 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 A002293 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 A002293 Ionut E. Iacob, T. Bruce McLean and Hua Wang, <a href="http://www.jstor.org/stable/10.4169/college.math.j.43.1.006">The V-flex, Triangle Orientation, and Catalan Numbers in Hexaflexagons</a>, The College Mathematics Journal, Vol. 43, No. 1 (January 2012), pp. 6-10.
%H A002293 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=286">Encyclopedia of Combinatorial Structures 286</a>.
%H A002293 V. A. Liskovets and T. R. Walsh, <a href="http://dx.doi.org/10.1016/j.aam.2005.03.006">Counting unrooted maps on the plane</a>, Advances in Applied Math., 36 No. 4 (2006), 364-387.
%H A002293 R. P. Loh, A. G. Shannon, and A. F. Horadam, <a href="/A000969/a000969.pdf">Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients</a>, preprint, 1980.
%H A002293 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=4).
%H A002293 Henri Muehle and Philippe Nadeau, <a href="https://arxiv.org/abs/1803.00540">A Poset Structure on the Alternating Group Generated by 3-Cycles</a>, arXiv:1803.00540 [math.CO], 2018.
%H A002293 J.-C. Novelli and 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 A002293 C. O. Oakley and R. J. Wisner, <a href="https://doi.org/10.2307/2310544">Flexagons</a>, Am. Math. Monthly 64 (3) (1957) 143-154, u_{3k+1}.
%H A002293 C. B. Pah and M. Saburov, <a href="http://dx.doi.org/10.5829/idosi.mejsr.2013.13.mae.9991">Single Polygon Counting on Cayley Tree of Order 4: Generalized Catalan Numbers</a>, Middle-East Journal of Scientific Research 13 (Mathematical Applications in Engineering): 01-05, 2013, ISSN 1990-9233.
%H A002293 Karol A. Penson and Karol Zyczkowski, <a href="http://dx.doi.org/10.1103/PhysRevE.83.061118">Product of Ginibre matrices : Fuss-Catalan and Raney distribution</a>, Phys. Rev. E 83, 061118, 15 June 2011.
%H A002293 Karol A. Penson and Karol Zyczkowski, <a href="http://arxiv.org/abs/1103.3453">Product of Ginibre matrices : Fuss-Catalan and Raney distribution</a>, arXiv:1103.3453 [math-ph], 2011.
%H A002293 Yuan (Friedrich) Qiu, Joe Sawada, and Aaron Williams, <a href="https://www.socs.uoguelph.ca/~sawada/papers/RSG.pdf">Maximize the Rightmost Digit: Gray Codes for Restricted Growth Strings</a>, WALCOM 2025. See p. 5.
%H A002293 Alison Schuetz and Gwyneth Whieldon, <a href="http://arxiv.org/abs/1401.7194">Polygonal Dissections and Reversions of Series</a>, arXiv:1401.7194 [math.CO], 2014.
%H A002293 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), Article 09.7.5.
%H A002293 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 A002293 Wikipedia, <a href="http://en.wikipedia.org/wiki/Fuss%E2%80%93Catalan_number">Fuss-Catalan number</a>
%H A002293 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 section 12.
%H A002293 Sophia Yakoubov, <a href="http://arxiv.org/abs/1310.2979">Pattern Avoidance in Extensions of Comb-Like Posets</a>, arXiv:1310.2979 [math.CO], 2013-2014.
%H A002293 Jun Yan, <a href="https://arxiv.org/abs/2501.01152">Lattice paths enumerations weighted by ascent lengths</a>, arXiv:2501.01152 [math.CO], 2025. See p. 7.
%H A002293 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.
%F A002293 O.g.f. satisfies: A(x) = 1 + x*A(x)^4 = 1/(1 - x*A(x)^3).
%F A002293 a(n) = binomial(4*n,n-1)/n, n >= 1, a(0) = 1. From the Lagrange series of the o.g.f. A(x) with its above given implicit equation.
%F A002293 From _Karol A. Penson_, Apr 02 2010: (Start)
%F A002293 Integral representation as n-th Hausdorff power moment of a positive function on the interval [0, 256/27]:
%F A002293 a(n) = Integral_{x=0..256/27}(x^n((3/256) * sqrt(2) * sqrt(3) * ((2/27) * 3^(3/4) * 27^(1/4) * 256^(/4) * hypergeom([-1/12, 1/4, 7/12], [1/2, 3/4], (27/256)*x)/(sqrt(Pi) * x^(3/4)) - (2/27) * sqrt(2) * sqrt(27) * sqrt(256) * hypergeom([1/6, 1/2, 5/6], [3/4, 5/4], (27/256)*x)/ (sqrt(Pi) * sqrt(x)) - (1/81) * 3^(1/4) * 27^(3/4) * 256^(1/4) * hypergeom([5/12, 3/4, 13/12], [5/4, 3/2], (27/256)*x/(sqrt(Pi)*x^(1/4)))/sqrt(Pi))).
%F A002293 This representation is unique as it represents the solution of the Hausdorff moment problem.
%F A002293 O.g.f.: hypergeom([1/4, 1/2, 3/4], [2/3, 4/3], (256/27)*x);
%F A002293 E.g.f.: hypergeom([1/4, 1/2, 3/4], [2/3, 1, 4/3], (256/27)*x). (End)
%F A002293 a(n) = upper left term in M^n, M = the production matrix:
%F A002293   1, 1
%F A002293   3, 3, 1
%F A002293   6, 6, 3, 1
%F A002293   ...
%F A002293 (where 1, 3, 6, 10, ...) is the triangular series. - _Gary W. Adamson_, Jul 08 2011
%F A002293 O.g.f. satisfies g = 1+x*g^4. If h is the series reversion of x*g, so h(x*g)=x, then (x-h(x))/x^2 is the o.g.f. of A006013. - _Mark van Hoeij_, Nov 10 2011
%F A002293 a(n) = binomial(4*n+1, n)/(4*n+1) = A062993(n+2,2). - _Robert FERREOL_, Apr 02 2015
%F A002293 a(n) = Sum_{i=0..n-1} Sum_{j=0..n-1-i} Sum_{k=0..n-1-i-j} a(i)*a(j)*a(k)*a(n-1-i-j-k) for n>=1; and a(0) = 1. - _Robert FERREOL_, Apr 02 2015
%F A002293 a(n) ~ 2^(8*n+1/2) / (sqrt(Pi) * n^(3/2) * 3^(3*n+3/2)). - _Vaclav Kotesovec_, Jun 03 2015
%F A002293 From _Peter Bala_, Oct 16 2015: (Start)
%F A002293 A(x)^2 is o.g.f. for A069271; A(x)^3 is o.g.f. for A006632;
%F A002293 A(x)^5 is o.g.f. for A196678; A(x)^6 is o.g.f. for A006633;
%F A002293 A(x)^7 is o.g.f. for A233658; A(x)^8 is o.g.f. for A233666;
%F A002293 A(x)^9 is o.g.f. for A006634; A(x)^10 is o.g.f. for A233667. (End)
%F A002293 D-finite with recurrence: a(n+1) = a(n)*4*(4*n + 3)*(4*n + 2)*(4*n + 1)/((3*n + 2)*(3*n + 3)*(3*n + 4)). - _Chai Wah Wu_, Feb 19 2016
%F A002293 E.g.f.: F([1/4, 1/2, 3/4], [2/3, 1, 4/3], 256*x/27), where F is the generalized hypergeometric function. - _Stefano Spezia_, Dec 27 2019
%F A002293 x*A'(x)/A(x) = (A(x) - 1)/(- 3*A(x) + 4) = x + 7*x^2 + 55*x^3 + 455*x^4 + ... is the o.g.f. of A224274. Cf. A001764 and A002294 - A002296. - _Peter Bala_, Feb 04 2022
%F A002293 a(n) = hypergeom([1 - n, -3*n], [2], 1). Row sums of A173020. - _Peter Bala_, Aug 31 2023
%F A002293 G.f.: t*exp(4*t*hypergeom([1, 1, 5/4, 3/2, 7/4], [4/3, 5/3, 2, 2], (256*t)/27))+1. - _Karol A. Penson_, Dec 20 2023
%F A002293 From _Karol A. Penson_, Jul 03 2024: (Start)
%F A002293 a(n) = Integral_{x=0..256/27} x^(n)*W(x)dx, n>=0, where W(x) = x^(-3/4) * sqrt(4*R(x) - 3^(3/4)*x^(1/4)/sqrt(R(x)))/(2*3^(1/4)*Pi), with R(x) = ((i + sqrt(3))*(4*sqrt(256 - 27*x) -12*i*sqrt(3*x))^(1/3))/16 - ((i - sqrt(3))*(4*sqrt(256 - 27*x) + 12*i* sqrt(3*x))^(1/3))/16, where i is the imaginary unit.
%F A002293 The elementary function W(x) is positive on the interval x = (0, 256/27) and is equal to the combination of hypergeometric functions in my formula from 2010; see above.
%F A002293 (Pi*W(x))^6 satisfies an algebraic equation of order 6, with integer polynomials as coefficients. (End)
%F A002293 G.f.: (Sum_{n >= 0} binomial(4*n+1, n)*x^n) / (Sum_{n >= 0} binomial(4*n, n)*x^n). - _Peter Bala_, Dec 14 2024
%F A002293 G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^7). - _Seiichi Manyama_, Jun 16 2025
%e A002293 There are a(2) = 4 quartic trees (vertex degree <= 4 and 4 possible branchings) with 2 vertices (one of them the root). Adding one more branch (one more vertex) to these four trees yields 4*4 + 6 = 22 = a(3) such trees.
%p A002293 series(RootOf(g = 1+x*g^4, g),x=0,20); # _Mark van Hoeij_, Nov 10 2011
%p A002293 seq(binomial(4*n, n)/(3*n+1),n=0..20); # _Robert FERREOL_, Apr 02 2015
%p A002293 # Using the integral representation above:
%p A002293 Digits:=6;
%p A002293 R:=proc(x)((I + sqrt(3))*(4*sqrt(256 - 27*x) - 12*I*sqrt(3)*sqrt(x))^(1/3))/16 - ((I - sqrt(3))*(4*sqrt(256 - 27*x) + 12*I*sqrt(3)*sqrt(x))^(1/3))/16;end;
%p A002293 W:=proc(x) x^(-3/4) * sqrt(4*R(x) - 3^(3/4)*x^(1/4)/sqrt(R(x)))/(2*3^(1/4)*Pi);end;
%p A002293 # Attention: W(x) is singular at x = 0. Integration is done from  a very small positive x to x = 256/27.
%p A002293 # For a(8):  ... gives 420732
%p A002293 evalf(int(x^8*W(x),x=0.000001..256/27));
%p A002293 # _Karol A. Penson_, Jul 05 2024
%t A002293 CoefficientList[InverseSeries[ Series[ y - y^4, {y, 0, 60}], x], x][[Range[2, 60, 3]]]
%t A002293 Table[Binomial[4n,n]/(3n+1),{n,0,25}] (* _Harvey P. Dale_, Apr 18 2011 *)
%t A002293 CoefficientList[1 + InverseSeries[Series[x/(1 + x)^4, {x, 0, 60}]], x] (* _Gheorghe Coserea_, Aug 12 2015 *)
%t A002293 terms = 22; A[_] = 0; Do[A[x_] = 1 + x*A[x]^4 + O[x]^terms, terms];
%t A002293 CoefficientList[A[x], x] (* _Jean-François Alcover_, Jan 13 2018 *)
%o A002293 (Magma) [ Binomial(4*n,n)/(3*n+1): n in [0..50]]; // _Vincenzo Librandi_, Apr 19 2011
%o A002293 (PARI) a(n)=binomial(4*n,n)/(3*n+1) /* _Charles R Greathouse IV_, Jun 16 2011 */
%o A002293 (PARI) my(x='x+O('x^33)); Vec(1 + serreverse(x/(1+x)^4)) \\ _Gheorghe Coserea_, Aug 12 2015
%o A002293 (Python)
%o A002293 A002293_list, x = [1], 1
%o A002293 for n in range(100):
%o A002293     x = x*4*(4*n+3)*(4*n+2)*(4*n+1)//((3*n+2)*(3*n+3)*(3*n+4))
%o A002293     A002293_list.append(x) # _Chai Wah Wu_, Feb 19 2016
%o A002293 (GAP) List([0..22],n->Binomial(4*n,n)/(3*n+1)); # _Muniru A Asiru_, Nov 01 2018
%Y A002293 Column k=3 of triangle A062993 and A070914.
%Y A002293 Cf. A000260, A002295, A002296, A027836, A062994, A346646 (binomial transform), A346664 (inverse binomial transform).
%Y A002293 Cf. A006632, A006633, A006634, A025174, A069271, A196678, A224274, A233658, A233666, A233667, A277877, A283049, A283101, A283102, A283103.
%Y A002293 Polyominoes: A005038 (oriented), A005040 (unoriented), A369471 (chiral), A369472 (achiral), A001764 {4,oo}, A002294 {6,oo}.
%Y A002293 Cf. A130564 (for generalized Catalan C(k, n), for = 4).
%K A002293 nonn,nice,easy
%O A002293 0,3
%A A002293 _N. J. A. Sloane_