A000139 a(n) = 2*(3*n)! / ((2*n+1)!*(n+1)!). (Formerly M1660 N0651)

2, 1, 2, 6, 22, 91, 408, 1938, 9614, 49335, 260130, 1402440, 7702632, 42975796, 243035536, 1390594458, 8038677054, 46892282815, 275750636070, 1633292229030, 9737153323590, 58392041019795, 352044769046880, 2132866978427640

Offset

0,1

Comments

This sequence arises in many different contexts, and over the years it has had several different definitions. I have now changed the definition back to one of the earlier ones, a self-contained formula. - N. J. A. Sloane, Apr 24 2012

The number of 2-stack sortable permutations on n letters (n >= 1).

The number of rooted non-separable planar maps with n+1 edges. - Valery A. Liskovets, Mar 17 2005

The shifted sequence starting with a(1): Number of quadrangular dissections of a square, counted by the number of vertices. Rooted, non-separable planar maps with no multiple edges, in which each non-root face has degree 4.

Number of left ternary trees having n nodes (n>=1). - Emeric Deutsch, Jul 23 2006

A combinatorial interpretation for this sequence in terms of a family of plane trees is given in [Schaeffer, Corollary 2 with k = 3]. - Peter Bala, Oct 12 2011

Number of canopy intervals in the Tamari lattices, see [Préville-Ratelle and Viennot, section 6]. - F. Chapoton, Apr 19 2015

The number of fighting fish (branching polyominoes). - David Bevan, Jan 10 2018

The number of 1324-avoiding dominoes (gridded permutations). - David Bevan, Jan 10 2018

For n > 0, a(n) is the number of simple strong triangulations of a fixed quadrilateral with n interior nodes. See A210664. - Andrew Howroyd, Feb 24 2021

References

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 365.

Eric S. Egge, Defying God: The Stanley-Wilf Conjecture, Stanley-Wilf Limits, and a Two-Generation Explosion of Combinatorics, pp. 65-82 of "A Century of Advancing Mathematics", ed. S. F. Kennedy et al., MAA Press 2015.

J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 714.

S. Kitaev, Patterns in Permutations and Words, Springer-Verlag, 2011. See p. 399 Table A.7

W. F. Lunnon, Counting polyominoes, pp. 347-372 of A. O. L. Atkin and B. J. Birch, editors, Computers in Number Theory. Academic Press, NY, 1971.

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).

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 6.41.

Links

T. D. Noe, Table of n, a(n) for n = 0..200

A. M. Baxter, Algorithms for Permutation Statistics, Ph. D. Dissertation, Rutgers University, May 2011. See p. 15.

E. Ben-Naim and P. L. Krapivsky, Popularity-Driven Networking, arXiv preprint arXiv:1112.0049 [cond-mat.stat-mech], 2011.

David Bevan, Robert Brignall, Andrew Elvey Price and Jay Pantone, A structural characterisation of Av(1324) and new bounds on its growth rate, arXiv preprint arXiv:1711.10325 [math.CO], 2017.

Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, A family of Bell transformations, arXiv:1803.07727 [math.CO], 2018.

Alin Bostan, Frédéric Chyzak, and Vincent Pilaud, Refined product formulas for Tamari intervals, arXiv:2303.10986 [math.CO], 2023.

M. Bousquet-Mélou and A. Jehanne, Polynomial equations with one catalytic variable, algebraic series and map enumeration, arXiv:math/0504018 [math.CO], 2005; J Comb. Thy. B 96 (2006), 623-672.

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

Colin Defant, Counting 3-Stack-Sortable Permutations, arXiv:1903.09138 [math.CO], 2019.

A. Del Lungo, F. Del Ristoro and J.-G. Penaud, Left ternary trees and non-separable rooted planar maps, Theor. Comp. Sci., 233, 2000, 201-215.

E. Duchi, V. Guerrini, S. Rinaldi and G. Schaeffer, Fighting fish: enumerative properties, Sém. Lothar. Combin. 78B (2017), Art. 43, 12 pp.

E. Duchi, V. Guerrini, S. Rinaldi, and G. Schaeffer, Fighting fish. J. Phys. A, Math. Theor. 50, No. 2, Article ID 024002, 16 p. (2017).

Enrica Duchi and Corentin Henriet, A bijection between rooted planar maps and generalized fighting fish, arXiv:2210.16635 [math.CO], 2022.

S. Dulucq, S. Gire, and O. Guibert, A combinatorial proof of J. West's conjecture Discrete Math. 187 (1998), no. 1-3, 71--96. MR1630680(99f:05053).

S. Dulucq, S. Gire, and J. West, Permutations with forbidden subsequences and nonseparable planar maps, Proceedings of the 5th Conference on Formal Power Series and Algebraic Combinatorics (Florence, 1993). Discrete Math. 153 (1996), no. 1-3, 85-103. MR1394948 (98a:05081)

Eric S. Egge, Defying God: The Stanley-Wilf Conjecture, Stanley-Wilf Limits, and a Two-Generation Explosion of Combinatorics, MAA Focus, August/September 2015, pp. 33-34. [Annotated scanned copy]

W. Fang, Fighting fish and two-stack sortable permutations, arXiv preprint arXiv:1711.05713 [math.CO], 2017.

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 713

I. Gessel and G. Xin, The generating function of ternary trees and continued fractions, Electron. J Combin. 13 (2006), paper 53.

Juan B. Gil, Oscar A. Lopez, and Michael D. Weiner, A positional statistic for 1324-avoiding permutations, arXiv:2311.18227 [math.CO], 2023.

O. Guibert, Stack words, standard Young tableaux, permutations with forbidden subsequences and planar maps, Discr. Math., 210 (2000), 71-85.

Elizabeth Hartung, Hung Phuc Hoang, Torsten Mütze, and Aaron Williams, Combinatorial generation via permutation languages. I. Fundamentals, arXiv:1906.06069 [cs.DM], 2019.

Hsien-Kuei Hwang, Mihyun Kang, and 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.

S. Kitaev, P. Salimov, C. Severs and H. Ulfarsson, Restricted non-separable planar maps and some pattern avoiding permutations, 2012.

S. Kitaev, P. Salimov, C. Severs and H. Ulfarsson, Restricted non-separable planar maps and some pattern avoiding permutations, Discrete Applied Mathematics, Volume 161, Issues 16-17, November 2013, Pages 2514-2526.

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

Sergey Kitaev, Pavel Salimov, Christopher Severs, and Henning Ulfarsson, Restricted rooted non-separable planar maps, arXiv preprint arXiv:1202.1790 [math.CO], 2012.

Arturo Merino and Torsten Mütze, Combinatorial generation via permutation languages. III. Rectangulations, arXiv:2103.09333 [math.CO], 2021.

Alois Panholzer, Parking function varieties for combinatorial tree models, arXiv:2007.14676 [math.CO], 2020.

L.-F. Préville-Ratelle and X. Viennot, An extension of Tamari lattices, arXiv preprint arXiv:1406.3787 [math.CO], 2014.

G. Schaeffer, A combinatorial interpretation of super-Catalan numbers of order two, (2001).

W. T. Tutte, A Census of Planar Maps, Canad. J. Math. 15 (1963), 249-271.

J. West, Sorting twice through a stack, Conference on Formal Power Series and Algebraic Combinatorics (Bordeaux, 1991), Theoret. Comput. Sci. 117 (1993), no. 1-2, 303-313.

D. Zeilberger, A proof of Julian West's conjecture that the number of two-stacksortable permutations of length n is 2(3n)!/((n + 1)!(2n + 1)!), Discrete Math., 102 (1992), 85-93.

P. Zinn-Justin and J.-B. Zuber, Matrix integrals and the counting of tangles and links, Discrete Math 246 (2002), 343-360.

P. Zinn-Justin and J.-B. Zuber, Matrix integrals and the generation and counting of virtual tangles and links, arXiv:math-ph/0303049, 2003; J. Knot Theor. Ramifications 13 (2004) 325-356.

Formula

a(n) = 2*binomial(3*n, 2*n+1)/(n*(n+1)), or 2*(3*n)!/((2*n+1)!*((n+1)!)).

Using Stirling's formula in A000142 it is easy to get the asymptotic expression a(n) ~ (27/4)^n / (sqrt(Pi*n / 3) * (2*n + 1) * (n + 1)). - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 13 2001

G.f.: A(z) = 2 + z*B(z), where B(z) = 1 - 8*z + 2*z*(5-6*z)*B - 2*z^2*(1+3*z)*B^2 - z^4*B^3.

G.f.: (2/(3*x)) * (hypergeom([-2/3, -1/3],[1/2],(27/4)*x)-1). - Mark van Hoeij, Nov 02 2009

G.f.: (2-3*R)/(R-1)^2 where R := RootOf(x-t*(t-1)^2,t) is an algebraic function in Maple notation. - Mark van Hoeij, Nov 08 2011

G.f.: 2*Q(0), where Q(k) = 1 + 3*x*(3*k+1)*(6*k+1)/(2*(k+1)*(4*k+3) - 6*x*(k+1)*(3*k+2)*(4*k+3)*(6*k+5)/(3*x*(3*k+2)*(6*k+5) + (2*k+3)*(4*k+5)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 30 2013

E.g.f.: 2*Q(0), where Q(k) = 1 + 3*x*(3*k+1)*(6*k+1)/(2*(k+1)*(2*k+1)*(4*k+3) - 6*x*(k+1)*(2*k+1)*(3*k+2)*(4*k+3)*(6*k+5)/(3*x*(3*k+2)*(6*k+5) + (2*k+2)*(2*k+3)*(4*k+5)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 30 2013

a(n) = A007318(3*n, 2*n+1)/A000217(n) for n > 0. - Reinhard Zumkeller, Feb 17 2013

a(n) is the n-th Hausdorff moment of the positive function w(x) defined on (0,27) which is equal (in Maple notation) to w(x) = 3*sqrt(3)*2^(2/3)*(3-sqrt(81-12*x)/9)*(1+sqrt(81-12*x)/9)^(1/3)/(8*Pi*x^(2/3))-sqrt(3)*2^(1/3)*(3+sqrt(81-12*x)/9)*(1+sqrt(81-12*x)/9)^(-1/3)/(4*Pi*x^(1/3)), that is, a(n) is the integral int(x^n * w(x), x=0..27/4), n=0,1,2,.... The function w(x) is unique. - Karol A. Penson, Jun 17 2013

D-finite with recurrence 2*(n+1)*(2*n+1)*a(n) -3*(3*n-1)*(3*n-2)*a(n-1)=0. - R. J. Mathar, Aug 21 2014

G.f. A(z) is related to the g.f. M(z) of A000168 by M(z) = 1 + A(z*M(z)^2) (see Tutte 1963, equation 6.3). - Noam Zeilberger, Nov 02 2016

From Ilya Gutkovskiy, Jan 17 2017: (Start)

E.g.f.: 2*2F2(1/3,2/3; 3/2,2; 27*x/4).

Sum_{n>=0} 1/a(n) = (1/2)*3F2(1,3/2,2; 1/3,2/3; 4/27) = 2.226206199291261... (End)

G.f. A(z) is the solution to the initial value problem 4*A + 2*z*A' = 8 + 3*z*A + 9*z^2*A' + 2*z^2*A*A', A(0) = 2. - Bjarki Ágúst Guðmundsson, Jul 03 2017

a(n+1) = a(n)*3*(3*n+1)*(3*n+2)/(2*(n+2)*(2*n+3)). - Chai Wah Wu, Apr 02 2021

a(n) = 4*(3*n)!/(n!*(2*n+2)!). - Chai Wah Wu, Dec 15 2021

From _Peter Bala, Feb 05 2022: (Start)

O.g.f.: A(x) = T(x)*(3 - T(x)), where T(x) = 1 + x*T(x)^3 is the o.g.f. of A001764.

(1/x)*(A(x) - 2)/(A(x) - 1) = 1 + x + 3*x^2 + 11*x^3 + 46*x^4 + 209*x^5 + ... is the o.g.f. of A233389.

1 + 2*x*A'(2*x)/A(2*x) = 1 + x + 7*x^2 + 61*x^3 + 591*x^4 + 6101*x^6 + ... is the o.g.f. of A218473.

Let B(x) = 1 + x*(A(x) - 1). Then x*B'(x)/B(x) = x + x^2 + 4*x^3 + 17*x^4 + 81*x^5 + ... is the o.g.f. of A121545. (End)

Example

G.f. = 2 + x + 2*x^2 + 6*x^3 + 22*x^4 + 91*x^5 + 408*x^6 + 1938*x^7 + ...

Maple

A000139 := n->2*(3*n)!/((2*n+1)!*((n+1)!)): seq(A000139(n), n=0..23);

Mathematica

Table[(2(3n)!)/((2n+1)!(n+1)!), {n, 0, 30}] (* Harvey P. Dale, Mar 31 2013 *)

Prog

(Haskell)
a000139 0 = 2
a000139 n = ((3 * n) `a007318` (2 * n + 1)) `div` a000217 n
-- Reinhard Zumkeller, Feb 17 2013
(Python)
from sympy import binomial
def A000139(n): return (binomial(3*n, n)*2)//((n+1)*(2*n+1))
(Sage)
def A000139(n): return (binomial(3*n, n)*2)//((n+1)*(2*n+1))
[A000139(n) for n in (0..23)]  # Peter Luschny, Jun 17 2013
(PARI) a(n)=binomial(3*n, n)*2/((n+1)*(2*n+1)); \\ Joerg Arndt, Jul 21 2014
(Magma) [2*Factorial(3*n)/(Factorial(2*n+1)*Factorial(n+1)): n in [0..25]]; // Vincenzo Librandi, Apr 20 2015
(Python)
A000139_list = [2]
for n in range(1, 30):
    A000139_list.append(3*(3*n-2)*(3*n-1)*A000139_list[-1]//(2*n+2)//(2*n+1)) # Chai Wah Wu, Apr 02 2021

Crossrefs

Cf. A000142, A000309, A001764, A005802, A006335, A004677, A121545, A218473, A233389.

Row m=1 of A210664(n>0).

Sequence in context: A032085 A032163 A038078 * A302190 A114572 A052621

Adjacent sequences: A000136 A000137 A000138 * A000140 A000141 A000142

Keyword

nonn,easy,nice,changed

Author

N. J. A. Sloane, entry revised Apr 24 2012

Status

approved