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A001764 a(n) = binomial(3*n,n)/(2*n+1) (enumerates ternary trees and also noncrossing trees).
(Formerly M2926 N1174)
1, 1, 3, 12, 55, 273, 1428, 7752, 43263, 246675, 1430715, 8414640, 50067108, 300830572, 1822766520, 11124755664, 68328754959, 422030545335, 2619631042665, 16332922290300, 102240109897695, 642312451217745, 4048514844039120, 25594403741131680, 162250238001816900 (list; graph; refs; listen; history; text; internal format)



Smallest number of straight line crossing-free spanning trees on n points in the plane.

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

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

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

Number of 12312-avoiding matchings on [2n].

Number of complete ternary trees with n internal nodes, or 3n edges.

Number of rooted plane trees with 2n edges, where every vertex has even outdegree ("even trees").

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

Pfaff-Fuss-Catalan sequence C^{m}_n for m=3, see the Graham et al. reference, p. 347. eq. 7.66.

Also 3-Raney sequence, see the Graham et al. reference, p. 346-7.

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

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

Central terms of pendular triangle A167763. - Philippe Deléham, Nov 12 2009

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

a(n) = A258708(2*n,n) for n > 0. - Reinhard Zumkeller, Jun 23 2015

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

Number of sum-indecomposable (4231,42513)-avoiding permutations. Conjecturally, number of sum-indecomposable (2431,45231)-avoiding permutations. - Alexander Burstein, Oct 19 2017

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

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

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

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

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

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

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

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


Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 23.

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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.

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.

W. Kuich, Languages and the enumeration of planted plane trees. Nederl. Akad. Wetensch. Proc. Ser. A 73 = Indag. Math. 32, (1970), 268-280.

T. V. Narayana, Lattice Path Combinatorics with Statistical Applications. Univ. Toronto Press, 1979, p. 98.

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.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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


G. C. Greubel, Table of n, a(n) for n = 0..1000 [Terms 0 to 100 computed by T. D. Noe; Terms 101 to 1000 by G. C. Greubel, Jan 13 2017]

V. E. Adler and A. B. Shabat, Volterra chain and Catalan numbers, arXiv:1810.13198 [nlin.SI], 2018.

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Joerg Arndt, Matters Computational (The Fxtbook), pp. 337-338.

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A. Asinowski, B. Hackl, and S. Selkirk, Down step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.

Jean-Christophe Aval, Multivariate Fuss-Catalan numbers, arXiv:0711.0906 [math.CO], 2007.

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I. Bajunaid et al., Function series, Catalan numbers and random walks on trees, Amer. Math. Monthly 112 (2005), 765-785.

P. Balduf, The propagator and diffeomorphisms of an interacting field theory, Master's thesis, submitted to the Institut für Physik, Mathematisch-Naturwissenschaftliche Fakultät, Humboldt-Universtität, Berlin, 2018.

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Paul Barry, Characterizations of the Borel triangle and Borel polynomials, arXiv:2001.08799 [math.CO], 2020.

Y. Baryshnikov, On Stokes sets, New developments in singularity theory (Cambridge, 2000): 65-86. Kluwer Acad. Publ., Dordrecht, 2001.

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Francois Bergeron, Combinatorics of r-Dyck paths, r-Parking functions, and the r-Tamari lattices, arXiv:1202.6269 [math.CO], 2012.

Olivier Bernardi and Nicolas Bonichon, Intervals in Catalan lattices and realizers of triangulations, Journal of Combinatorial Theory, Series A 116:1 (2009), pp. 55-75. See also Bernardi's slides, Catalan lattices and realizers of triangulations (April 2007).

D. Bevan, D. Levin, P. Nugent, J. Pantone, and L. Pudwell, Pattern avoidance in forests of binary shrubs, arXiv:1510:08036 [math.CO], 2015.

D. Birmajer, J. B. Gil and M. D. Weiner, Colored partitions of a convex polygon by noncrossing diagonals, arXiv:1503.05242 [math.CO], 2015.

Michel Bousquet and Cédric Lamathe, On symmetric structures of order two, Discrete Math. Theor. Comput. Sci. 10 (2008), 153-176.

M. Bousquet-Mélou and M. Petkovšek, Walks confined in a quadrant are not always D-finite, arXiv:math/0211432 [math.CO], 2002.

Włodzimierz Bryc, Raouf Fakhfakh, and Wojciech Młotkowski, Cauchy-Stieltjes families with polynomial variance functions and generalized orthogonality, arXiv:1708.05343 [math.PR], 2017-2019. Also in Probability and Mathematical Statistics 39(2) (2019), 237-258.

N. T. Cameron, Random walks, trees and extensions of Riordan group techniques, Dissertation, Howard University, 2002.

Naiomi Cameron and J. E. McLeod, Returns and Hills on Generalized Dyck Paths, Journal of Integer Sequences, Vol. 19, 2016, #16.6.1.

Peter J. Cameron and Liam Stott, Trees and cycles, arXiv:2010.14902 [math.CO], 2020. See p. 33.

L. Carlitz, Enumeration of two-line arrays, Fib. Quart., 11(2) (1973), 113-130.

F. Cazals, Combinatorics of Non-Crossing Configurations, Studies in Automatic Combinatorics, Volume II (1997).

Matteo Cervetti and Luca Ferrari, Pattern avoidance in the matching pattern poset, arXiv:2009.01024 [math.CO], 2020.

W. Y. C. Chen, T. Mansour and S. H. F. Yan, Matchings avoiding partial patterns, arXiv:math/0504342 [math.CO], 2005.

J. Cigler, Some remarks about q-Chebyshev polynomials and q-Catalan numbers and related results, arXiv:1312.2767 [math.CO], 2013.

B. Collins, I. Nechita, and K. Zyczkowski, Random graph states, maximal flow and Fuss-Catalan distributions, arXiv:1003.3075 [quant-ph], 2010.

T. C. Copeland, Compositional inverse pairs, the Burgers-Hopf equation, and the Stasheff associahedra, 2014.

T. C. Copeland, Discriminating Deltas, Depressed Equations, and Generalized Catalan Numbers, 2012.

S. J. Cyvin, Jianji Wang, J. Brunvoll, Shiming Cao, Ying Li, B. N. Cyvin, and Yugang Wang, Staggered conformers of alkanes: complete solution of the enumeration problem, J. Molec. Struct. 413-414 (1997), 227-239.

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Colin Defant, Catalan Intervals and Uniquely Sorted Permutations, arXiv:1904.02627 [math.CO], 2019.

C. Defant and N. Kravitz, Stack-sorting for words, arXiv:1809.09158 [math.CO], 2018.

E. Deutsch, S. Feretic and M. Noy, Diagonally convex directed polyominoes and even trees: a bijection and related issues, Discrete Math., 256 (2002), 645-654.

S. Dulucq, Etude combinatoire de problèmes d'énumération, d'algorithmique sur les arbres et de codage par des mots, a thesis presented to l'Université de Bordeaux I, 1987. (Annotated scanned copy)

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R. Dickau, Fuss-Catalan Numbers. Figures of various interpretations.

C. Domb and A. J. Barrett, Enumeration of ladder graphs, Discrete Math. 9 (1974), 341-358.

C. Domb and A. J. Barrett, Enumeration of ladder graphs, Discrete Math. 9 (1974), 341-358. (Annotated scanned copy)

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J. A. Eidswick, Short factorizations of permutations into transpositions, Disc. Math. 73 (1989) 239-243

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I. M. H. Etherington, Some problems of non-associative combinations, Edinburgh Math. Notes, 32 (1940), 1-6.

I. M. H. Etherington, Some problems of non-associative combinations (I), 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.

Jishe Feng, The Hessenberg matrices and Catalan and its generalized numbers, arXiv:1810.09170 [math.CO], 2018. See p. 4.

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 486.

N. Gabriel, K. Peske, L. Pudwell, and S. Tay, Pattern Avoidance in Ternary Trees, J. Int. Seq. 15 (2012), #12.1.5

I. Gessel and G. Xin, The generating function of ternary trees and continued fractions, arXiv:math/0505217 [math.CO], 2005.

S. Goldstein, J. L. Lebowitz, E. R. Speer, The Discrete-Time Facilitated Totally Asymmetric Simple Exclusion Process, arXiv:2003.04995 [math-ph], 2020.

N. S. S. Gu, N. Y. Li and T. Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.

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T.-X. He, L. W. Shapiro, Fuss-Catalan matrices, their weighted sums, and stabilizer subgroups of the Riordan group, Lin. Alg. Applic. 532 (2017) 25-41, Fuss-Catalan Number (F_3)_n

V. E. Hoggatt, Jr., Letters to N. J. A. Sloane, 1974-1975.

V. E. Hoggatt, Jr., 7-page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977.

V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., 14 (1976), 395-405.

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INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 53.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 285.

C. Ji and J. Propp, Brussels Sprouts, Noncrossing Trees, and Parking Functions, arXiv:1805.03608 [math.CO], 2018.

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S. Kitaev and A. de Mier, Enumeration of fixed points of an involution on beta(1, 0)-trees, arXiv:1210.2618 [math.CO], 2012.

Don Knuth, 20th Anniversary Christmas Tree Lecture.

G. Kreweras, Sur les partitions non croisées d'un cycle, (French) Discrete Math. 1(4) (1972), 333-350. MR0309747 (46 #8852).

D. V. Kruchinin, On solving some functional equations, Advances in Difference Equations (2015), 2015:17.

Dmitry V. Kruchinin and Vladimir V. Kruchinin, A Generating Function for the Diagonal T_{2n,n} in Triangles, Journal of Integer Sequences, 18 (2015), Article 15.4.6.

Markus Kuba and Alois Panholzer, Enumeration formulas for pattern restricted Stirling permutations, Discrete Math. 312(21) (2012), 3179--3194. MR2957938. - From N. J. A. Sloane, Sep 25 2012

Woldieter Lang, Ternary trees with n = 1, 2, 3 and 4 vertices.

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R. P. Loh, A. G. Shannon, and A. F. Horadam, Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients, preprint, 1980.

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W. Mlotkowski, M. Nowak, K. Penson, and K. Zyczkowski, Spectral density of generalized Wishart matrices and free multiplicative convolution, arXiv preprint arXiv:1407.1282 [math-ph], 2015.

W. Mlotkowski and K. A. Penson, The probability measure corresponding to 2-plane trees, arXiv:1304.6544 [math.PR], 2013.

Hanna Mularczyk, Lattice Paths and Pattern-Avoiding Uniquely Sorted Permutations, arXiv:1908.04025 [math.CO], 2019.

Emanuele Munarini, Shifting Property for Riordan, Sheffer and Connection Constants Matrices, Journal of Integer Sequences, 20 (2017), Article 17.8.2.

H. Niederhausen, Catalan Traffic at the Beach, Electronic Journal of Combinatorics, 9 (2002), #R33.

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Helmut Prodinger, Generating functions for a lattice path model introduced by Deutsch, arXiv:2004.04215 [math.CO], 2020.

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Index entries for "core" sequences

Index entries for sequences related to trees


From Karol A. Penson, Nov 08 2001: (Start)

G.f.: (2/sqrt(3*x))*sin((1/3)*arcsin(sqrt(27*x/4))).

E.g.f.: hypergeom([1/3, 2/3], [1, 3/2], 27/4*x).

Integral representation as n-th moment of a positive function on [0, 27/4]: a(n) = Integral_{x=0..6.75} (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)

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

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).

G.f. Rev(x/c(x))/x, where c(x) is the g.f. of A000108 (Rev=reversion of). - Paul Barry, Mar 26 2010

From Gary W. Adamson, Jul 07 2011: (Start)

Let M = the production matrix:

  1, 1

  2, 2, 1

  3, 3, 2, 1

  4, 4, 3, 2, 1

  5, 5, 4, 3, 2, 1


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)

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

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

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

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

(n + 1) * a(n) = A174687(n).

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

a(n) = binomial(3*n+1, n)/(3*n+1) = A062993(n+1,1). - Robert FERREOL, Apr 03 2015

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

a(n) ~ 3^(3*n + 1/2)/(sqrt(Pi)*4^(n+1)*n^(3/2)). - Ilya Gutkovskiy, Nov 21 2016

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

From Peter Bala, Sep 14 2021: (Start)

A(x) = exp( Sum_{n >= 1} (1/3)*binomial(3*n,n)*x^n/n ).

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)

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

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


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.

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 + ...


A001764 := n->binomial(3*n, n)/(2*n+1): seq(A001764(n), n=0..25);

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

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

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

alias(PS=ListTools:-PartialSums): A001764List := proc(m) local A, P, n;

A := [1, 1]; P := [1]; for n from 1 to m - 2 do P := PS(PS([op(P), P[-1]]));

A := [op(A), P[-1]] od; A end: A001764List(25); # Peter Luschny, Mar 26 2022


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 *)

Table[Binomial[3n, n]/(2n+1), {n, 0, 25}] (* Harvey P. Dale, Jul 24 2011 *)


(PARI) {a(n) = if( n<0, 0, (3*n)! / n! / (2*n + 1)!)};

(PARI) {a(n) = if( n<0, 0, polcoeff( serreverse( x - x^3 + O(x^(2*n + 2))), 2*n + 1))};

(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))};

(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

(PARI) Vec(1 + serreverse(x / (1+x)^3 + O(x^30))) \\ Gheorghe Coserea, Aug 05 2015


def A001764_list(n) :

    D = [0]*(n+1); D[1] = 1

    R = []; b = false; h = 1

    for i in range(2*n) :

        for k in (1..h) : D[k] += D[k-1]

        if not b : R.append(D[h])

        else : h += 1

        b = not b

    return R

A001764_list(22) # Peter Luschny, May 03 2012

(MAGMA) [Binomial(3*n, n)/(2*n+1): n in [0..30]]; // Vincenzo Librandi, Sep 04 2014


a001764 n = a001764_list !! n

a001764_list = 1 : [a258708 (2 * n) n | n <- [1..]]

-- Reinhard Zumkeller, Jun 23 2015

(GAP) List([0..25], n->Binomial(3*n, n)/(2*n+1)); # Muniru A Asiru, Oct 31 2018


Cf. A000108, A001762, A001763, A002293 - A002296, A006013, A025174, A063548, A064017, A072247, A072248, A134264, A143603, A258708, A256311.

A column of triangle A102537.

Bisection of A047749 and A047761.

Row sums of triangle A108410.

Second column of triangle A062993.

Mod 3 = A113047.

Sequence in context: A024038 A007199 A179848 * A171780 A216493 A216494

Adjacent sequences:  A001761 A001762 A001763 * A001765 A001766 A001767




N. J. A. Sloane



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Last modified May 23 23:53 EDT 2022. Contains 353993 sequences. (Running on oeis4.)