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A006013 a(n) = binomial(3*n+1,n)/(n+1).
(Formerly M1782)
73
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)
OFFSET

0,2

COMMENTS

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

REFERENCES

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

LINKS

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..

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.

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.

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

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.

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.

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.

FORMULA

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)

EXAMPLE

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

MAPLE

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

MATHEMATICA

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 *)

PROG

(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))

(Sage)

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

(Haskell)

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

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

-- Reinhard Zumkeller, Jun 22 2015

CROSSREFS

Cf. A121645, A115728, A143603, A236194.

These are the odd indices of A047749.

Cf. A007318, A258708.

Sequence in context: A260773 A174796 A046648 * A187979 A243632 A196148

Adjacent sequences:  A006010 A006011 A006012 * A006014 A006015 A006016

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

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Last modified November 20 10:41 EST 2017. Contains 294963 sequences.