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A006013 a(n) = binomial(3*n+1,n)/(n+1).
(Formerly M1782)
1, 2, 7, 30, 143, 728, 3876, 21318, 120175, 690690, 4032015, 23841480, 142498692, 859515920, 5225264024, 31983672534, 196947587823, 1219199353190, 7583142491925, 47365474641870, 296983176369495, 1868545312633440, 11793499763070480 (list; graph; refs; listen; history; text; internal format)



Enumerates pairs of ternary trees [Knuth, 2014]. - N. J. A. Sloane, Dec 09 2014

G.f. (offset 1) is series reversion of x - 2x^2 + x^3.

Hankel transform is A005156(n+1). - Paul Barry, Jan 20 2007

a(n) = number of ways to connect 2n-2 points labeled 1,2,...,2n-2 in a line with 0 or more noncrossing arcs above the line such that each maximal contiguous sequence of isolated points has even length. For example, with arcs separated by dashes, a(3)=7 counts {} (no arcs), 12, 14, 23, 34, 12-34, 14-23. It does not count 13 because 2 is an isolated point. - David Callan, Sep 18 2007

In my 2003 paper I introduced L-algebras. These are K-vector spaces equipped with two binary operations > and < satisfying (x>y)<z = x>(y<z). In my arXiv paper math-ph/0709.3453 I show that the free L-algebra on one generator is related to symmetric ternary trees with odd degrees, so the dimensions of the homogeneous components are 1,2,7,30,143,.... These L-algebras are closely related to the so-called triplicial-algebras, 3 associative operations and 3 relations whose free object is related to even trees. - Philippe Leroux (ph_ler_math(AT)yahoo.com), Oct 05 2007

a(n-1) is also the number of projective dependency trees with n nodes. [Marco Kuhlmann (marco.kuhlmann(AT)lingfil.uu.se), Apr 06 2010]

Number of subpartitions of [1^2,2^2,...,n^2]. - Franklin T. Adams-Watters, Apr 13 2011

a(n) = sum of (n+1)-th row terms of triangle A143603. - Gary W. Adamson, Jul 07 2011

Also the number of Dyck n-paths with up steps colored in two ways (N or A) and avoiding NA. The 7 Dyck 2-paths are NDND, ADND, NDAD, ADAD, NNDD, ANDD, and AADD. - David Scambler, Jun 24 2013

a(n) is also the number of permutations avoiding 213 in the classical sense which can be realized as labels on an increasing strict binary tree with 2n-1 nodes. See A245904 for more information on increasing strict binary trees. - Manda Riehl Aug 07 2014

With offset 1, a(n) is the number of ordered trees (A000108) with n non-leaf vertices and n leaf vertices such that every non-leaf vertex has a leaf child (and hence exactly one leaf child). - David Callan, Aug 20 2014

a(n) = A258708(2*n+1,n). - Reinhard Zumkeller, Jun 22 2015

a(n) = A110616(n+1,1). - Ira M. Gessel, Jan 04 2018

a(n) is the number of paths in the plane with unit east and north steps, never going above the line x=2y, from (0,0) to (2n+1,n). - Ira M. Gessel, Jan 04 2018

a(n) is the number of words on the alphabet [n+1] that avoid the patterns 231 and 221 and contain exactly one 1 and exactly two occurrences of every other letter. - Colin Defant, Sep 26 2018

a(n) is the number of Motzkin paths of length 3n with n of each type of step, such that (1, 1) and (1, 0) alternate (ignoring (-1, 1) steps). All paths start with a (1, 1) step. - Helmut Prodinger, Apr 08 2019


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 14 2017]

A. Aggarwal, Armstrong's Conjecture for (k, mk+1)-Core Partitions, arXiv preprint arXiv:1407.5134 [math.CO], 2014.

C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps, FPSAC02, Melbourne, 2002.

Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.

W. G. Brown, Enumeration of non-separable planar maps, Canad. J. Math., 15 (1963), 526-545.

W. G. Brown, Enumeration of non-separable planar maps [Annotated scanned copy]

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

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

F. Chapoton, F. Hivert, J.-C. Novelli, A set-operad of formal fractions and dendriform-like sub-operads, arXiv preprint arXiv:1307.0092 [math.CO], 2013.

F. Chapoton, S. Giraudo, Enveloping operads and bicoloured noncrossing configurations, arXiv preprint arXiv:1310.4521 [math.CO], 2013.

Jins de Jong, Alexander Hock, Raimar Wulkenhaar, Catalan tables and a recursion relation in noncommutative quantum field theory, arXiv:1904.11231 [math-ph], 2019.

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

Isaac DeJager, Madeleine Naquin, Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.

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.

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

Samuele Giraudo, Tree series and pattern avoidance in syntax trees, arXiv:1903.00677 [math.CO], 2019.

Hsien-Kuei Hwang, Mihyun Kang, Guan-Huei Duh, Asymptotic Expansions for Sub-Critical Lagrangean Forms, LIPIcs Proceedings of Analysis of Algorithms 2018, Vol. 110. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2018.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 432

S. Kitaev and A. de Mier, Enumeration of fixed points of an involution on beta(1, 0)-trees, arXiv preprint arXiv:1210.2618 [math.CO], 2012. - From N. J. A. Sloane, Dec 31 2012

Sergey Kitaev, Anna de Mier, Marc Noy, On the number of self-dual rooted maps, European J. Combin. 35 (2014), 377--387. MR3090510. See Theorem 1. - N. J. A. Sloane, May 19 2014

Don Knuth, 20th Anniversary Christmas Tree Lecture

Philippe Leroux, An algebraic framework of weighted directed graphs, Int. J. Math. Math. Sci. 58. (2003).

Philippe Leroux, L-algebras, triplicial-algebras, within an equivalence of categories motivated by graphs, arXiv:0709.3453 [math-ph], 2008.

Ho-Hon Leung, Thotsaporn "Aek" Thanatipanonda, A Probabilistic Two-Pile Game, arXiv:1903.03274 [math.CO], 2019.

Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.

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

Henri Muehle, Philippe Nadeau, A Poset Structure on the Alternating Group Generated by 3-Cycles, arXiv:1803.00540 [math.CO], 2018.

Liviu I. Nicolaescu, Counting Morse functions on the 2-sphere, arXiv:math/0512496 [math.GT], 2005.

J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv:1403.5962 [math.CO], 2014.

M. Noy, Enumeration of noncrossing trees on a circle, Discrete Math., 180, 301-313, 1998.

J.-B. Priez, A. Virmaux, Non-commutative Frobenius characteristic of generalized parking functions: Application to enumeration, arXiv:1411.4161 [math.CO], 2014-2015.

Helmut Prodinger, On some questions by Cameron about ternary paths --- a linear algebra approach, arXiv:1910.02320 [math.CO], 2019.

Helmut Prodinger, Sarah J. Selkirk, Stephan Wagner, On two subclasses of Motzkin paths and their relation to ternary trees, arXiv:1902.01681 [math.CO], 2019.

Jocelyn Quaintance, Combinatoric Enumeration of Two-Dimensional Proper Arrays, Discrete Math., 307 (2007), 1844-1864.

Thomas M. Richardson, The three 'R's and Dual Riordan Arrays, arXiv:1609.01193 [math.CO], 2016.

D. G. Rogers, Comments on A111160, A055113 and A006013

M. Somos, Number Walls in Combinatorics.

Zhujun Zhang, A Note on Counting Dependency Trees, arXiv:1708.08789 [math.GM], 2017. See p. 3.

S.-n. Zheng and S.-l. Yang, On the-Shifted Central Coefficients of Riordan Matrices, Journal of Applied Mathematics, Volume 2014, Article ID 848374, 8 pages.


G.f. is square of g.f. for ternary trees, A001764 [Knuth, 2014]. - N. J. A. Sloane, Dec 09 2014

Convolution of A001764 with itself: 2*C(3*n+2,n)/(3*n+2), or C(3*n+2,n+1)/(3*n+2).

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

G.f.: A(x)/x with A(x)=x/(1-A(x))^2. - Vladimir Kruchinin, Dec 26 2010

From Gary W. Adamson, Jul 14 2011: (Start)

a(n) is the top left term in M^n, where M is the infinite square production matrix:

2, 1, 0, 0, 0, ...

3, 2, 1, 0, 0, ...

4, 3, 2, 1, 0, ...

5, 4, 3, 2, 1, ...

... (End)

From Gary W. Adamson, Aug 11 2011: (Start)

a(n) is the sum of top row terms in Q^n, where Q is the infinite square production matrix as follows:

1, 1, 0, 0, 0, ...

2, 2, 1, 0, 0, ...

3, 3, 2, 1, 0, ...

4, 4, 3, 2, 1, ...

... (End)

2*(n+1)*(2n+1)*a(n) = 3*(3n-1)*(3n+1)*a(n-1). - R. J. Mathar, Dec 17 2011

a(n) = 2*A236194(n)/n for n>0. - Bruno Berselli, Jan 20 2014

From Ilya Gutkovskiy, Dec 29 2016: (Start)

E.g.f.: 2F2(2/3,4/3; 3/2,2; 27*x/4).

a(n) ~ 3^(3*n+3/2)/(sqrt(Pi)*4^(n+1)*n^(3/2)). (End)


a(3) = 30 since the top row of Q^3 = (12, 12, 5, 1).


BB:=[T, {T=Prod(Z, Z, F, F), F=Sequence(B), B=Prod(F, F, Z)}, unlabeled]: seq(count(BB, size=i), i=2..24); # Zerinvary Lajos, Apr 22 2007


InverseSeries[Series[y-2*y^2+y^3, {y, 0, 32}], x]

Binomial[3#+1, #]/(#+1)&/@Range[0, 30]  (* Harvey P. Dale, Mar 16 2011 *)


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

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


def A006013_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 b : R.append(D[h]); h += 1

        b = not b

    return R

A006013_list(23) # Peter Luschny, May 03 2012

(MAGMA) [Binomial(3*n+1, n)/(n+1): n in [0..30]]; // Vincenzo Librandi, Mar 29 2015


a006013 n = a007318 (3 * n + 1) n `div` (n + 1)

a006013' n = a258708 (2 * n + 1) n

-- Reinhard Zumkeller, Jun 22 2015


These are the odd indices of A047749.

Cf. A121645, A115728, A143603, A236194.

Cf. A007318, A071948, A110616, A258708.

Sequence in context: A260773 A174796 A046648 * A187979 A243632 A196148

Adjacent sequences:  A006010 A006011 A006012 * A006014 A006015 A006016




N. J. A. Sloane


Edited by N. J. A. Sloane, Feb 21 2008



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Last modified January 18 16:27 EST 2020. Contains 331011 sequences. (Running on oeis4.)