A006335 -id:A006335 - cp's OEIS frontend

A000139 a(n) = 2(3n)! / ((2n+1)!(n+1)!).

2, 1, 2, 6, 22, 91, 408, 1938, 9614, 49335, 260130, 1402440, 7702632, 42975796, 243035536, 1390594458, 8038677054, 46892282815, 275750636070, 1633292229030, 9737153323590, 58392041019795, 352044769046880, 2132866978427640, 12980019040145352, 79319075627675556

Offset: 0

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

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
Conjecture: a(n) is odd iff n is a term of A022341. - Peter Bala, Jul 24 2025

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

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.

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.
Alin Bostan, Frédéric Chyzak, Bérénice Delcroix-Oger, Guillaume Laplante-Anfossi, Vincent Pilaud, and Kurt Stoeckl, Diagonals of permutahedra and associahedra, Sém. Lotharingien Comb., 37th Formal Power Series Alg. Comb. (FPSAC 2025). See pp. 10-11.
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 non-separable planar maps and some pattern avoiding permutations, arXiv preprint arXiv:1202.1790 [math.CO], 2012. [Original title: Restricted rooted non-separable planar maps]
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.
Permutation Pattern Avoidance Library (PermPAL), 1324 Domino
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.

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

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

a000139 0 = 2
a000139 n = ((3 * n) `a007318` (2 * n + 1)) `div` a000217 n
-- Reinhard Zumkeller, Feb 17 2013

[2*Factorial(3*n)/(Factorial(2*n+1)*Factorial(n+1)): n in [0..25]]; // Vincenzo Librandi, Apr 20 2015

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

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

a(n)=binomial(3*n,n)*2/((n+1)*(2*n+1)); \\ Joerg Arndt, Jul 21 2014

from sympy import binomial
def A000139(n): return (binomial(3*n, n)*2)//((n+1)*(2*n+1))

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

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

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 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 Integral_{x=0..27/4} x^n*w(x) dx, n >= 0. 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)

A000309 Number of rooted planar bridgeless cubic maps with 2n nodes.

Original entry on oeis.org

1, 1, 4, 24, 176, 1456, 13056, 124032, 1230592, 12629760, 133186560, 1436098560, 15774990336, 176028860416, 1990947110912, 22783499599872, 263411369705472, 3073132646563840, 36143187370967040, 428157758086840320, 5105072641718353920, 61228492804372561920

Offset: 0

N. J. A. Sloane and Robert G. Wilson v

Also counts rooted planar non-separable triangulations with 3n edges. - Valery A. Liskovets, Dec 01 2003
Equivalently, rooted planar loopless triangulations with 2n triangles. - Noam Zeilberger, Oct 06 2016
Description trees of type (2,2) with n edges. (A description tree of type (a,b) is a rooted plane tree where every internal node is labeled by an integer between a and [b + sum of labels of its children], every leaf is labeled a, and the root is labeled [b + sum of labels of its children]. See Definition 1 and Section 5.2 of Cori and Schaeffer 2003.) - Noam Zeilberger, Oct 08 2017
The September 2018 talk by Noam Zeilberger (see link to video) connects three topics (planar maps, Tamari lattices, lambda calculus) and eight sequences: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827. - N. J. A. Sloane, Sep 17 2018

C. F. Earl and L. J. March, Architectural applications of graph theory, pp. 327-355 of R. J. Wilson and L. W. Beineke, editors, Applications of Graph Theory. Academic Press, NY, 1979.
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).

T. D. Noe, Table of n, a(n) for n = 0..100
Marie Albenque, Dominique Poulalhon, A Generic Method for Bijections between Blossoming Trees and Planar Maps, Electron. J. Combin., 22 (2015), #P2.38.
Dario Benedetti, Sylvain Carrozza, Reiko Toriumi, Guillaume Valette, Multiple scaling limits of U(N)^2 X O(D) multi-matrix models, arXiv:2003.02100 [math-ph], 2020.
Olivier Bernardi, Bijective counting of Kreweras walks and loopless triangulations, Journal of Combinatorial Theory, Series A 114:5 (2007), 931-956.
Junliang Cai, Yanpei Liu, The enumeration of rooted nonseparable nearly cubic maps, Discrete Math. 207 (1999), no. 1-3, 9--24. MR1710479 (2000g:05074). See (31).
Robert Cori and Gilles Schaeffer, Description trees and Tutte formulas, Theoretical Computer Science 292:1 (2003), 165-183.
S. Dulucq and O. Guibert, Stack words, standard tableaux and Baxter permutations, Disc. Math. 157 (1996), 91-106.
C. F. Earl and L. J. March, Architectural applications of graph theory, pp. 327-355 of R. J. Wilson and L. W. Beineke, editors, Applications of Graph Theory. Academic Press, NY, 1979. (Annotated scanned copy)
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.
R. C. Mullin, On counting rooted triangular maps, Canad. J. Math., v.17 (1965), 373-382.
Elena Patyukova, Taylor Rottreau, Robert Evans, Paul D. Topham, Martin J. Greenall, Hydrogen Bonding Aggregation in Acrylamide: Theory and Experiment, Macromolecules (2018) Vol. 51, No. 18, 7032-7043. Also arXiv:1805.09878 [math.CA], 2018.
W. T. Tutte, A census of Hamiltonian polygons, Canad. J. Math., 14 (1962), 402-417.
W. T. Tutte, On the enumeration of four-colored maps, SIAM J. Appl. Math., 17 (1969), 454-460.
Noam Zeilberger, A theory of linear typings as flows on 3-valent graphs, arXiv:1804.10540 [cs.LO], 2018.
Noam Zeilberger, A Sequent Calculus for a Semi-Associative Law, arXiv:1803.10080 [math.LO], March 2018 (A revised version of a 2017 conference paper)
Noam Zeilberger, A proof-theoretic analysis of the rotation lattice of binary trees, Part 1 (video), Part 2, Rutgers Experimental Math Seminar, Sep 13 2018.
Jian Zhou, Fat and Thin Emergent Geometries of Hermitian One-Matrix Models, arXiv:1810.03883 [math-ph], 2018.

Sequences mentioned in the Noam Zeilberger 2018 video: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827.

Cf. A000139, A000264, A000356, A002005, A006335, A358091.

List([0..20], n -> 2^(n+1)*Factorial(3*n)/(Factorial(n)* Factorial(2*n+2))); # G. C. Greubel, Nov 29 2018

[2^(n+1)*Factorial(3*n)/(Factorial(n)*Factorial(2*n+2)): n in [0..20]]; // Vincenzo Librandi, Aug 10 2014

a := n -> 2^(n+1)*(3*n)!/(n!*(2*n+2)!);
A000309 := n -> -(-2)^(n-1)*(3*n+2)*hypergeom([-3*(n+1),-n,-n+1/3], [-n-1,-n-2/3], 1): seq(simplify(A000309(n)), n = 0..21); # Peter Luschny, Oct 28 2022

f[n_] := 2^n(3n)!/((n + 1)!(2n + 1)!); Table[f[n], {n, 0, 19}] (* Robert G. Wilson v, Sep 21 2004 *)
Join[{1},RecurrenceTable[{a[1]==1,a[n]==4a[n-1] Binomial[3n,3]/ Binomial[2n+2,3]}, a[n],{n,20}]] (* Harvey P. Dale, May 11 2011 *)

a(n) = 2^(n+1)*(3*n)!/(n!*(2*n+2)!); \\ Michel Marcus, Aug 09 2014

[2^n*factorial(3*n)/(factorial(n+1)*factorial(2*n+1))for n in range(20)] # G. C. Greubel Nov 29 2018

a(n) = 2^(n-1) * A000139(n) for n > 0.
a(n) = 4*a(n-1)*binomial(3*n, 3) / binomial(2*n+2, 3).
a(n) = 2^n*(3*n)!/ ( (n+1)!*(2*n+1)! ).
G.f.: (1/(6*x)) * (hypergeom([ -2/3, -1/3],[1/2],(27/2)*x)-1). - Mark van Hoeij, Nov 02 2009
a(n) ~ 3^(3*n+1/2)/(sqrt(Pi)*2^(n+2)*n^(5/2)). - Ilya Gutkovskiy, Oct 06 2016
D-finite with recurrence (n+1)*(2*n+1)*a(n) -3*(3*n-1)*(3*n-2)*a(n-1)=0. - R. J. Mathar, Nov 02 2018
a(n) = -(-2)^(n-1)*(3*n+2)*hypergeom([-3*(n+1),-n,-n+1/3], [-n-1,-n-2/3], 1). The a(n) are values of the polynomials A358091. - Peter Luschny, Oct 28 2022
From Karol A. Penson, Feb 24 2025: (Start)
G.f.: hypergeom([1/3, 2/3, 1], [3/2, 2], (27*z)/2).
G.f. A(z) satisfies: - 1 + 27*z + (-36*z + 1)*A(z) + 8*z*A(z)^2 + 16*z^2*A(z)^3 = 0.
G.f.: ((4*sqrt(4 - 54*z) + 12*i*sqrt(6)*sqrt(z))^(1/3)*(sqrt(z*(4 - 54*z)) - 9*i*sqrt(6)*z) + (4*sqrt(4 - 54*z) - 12*i*sqrt(6)*sqrt(z))^(1/3)*(9*i*sqrt(6)*z + sqrt(z*(4 - 54*z))) - 8*sqrt(z))/(48*z^(3/2)), where i = sqrt(-1) is the imaginary unit.
a(n) = Integral_{x=0..27/2} x^n*W(x), where W(x) = (6^(1/3)*(9 + sqrt(81 - 6*x))^(2/3)*(9*sqrt(3) - sqrt(27 - 2*x)) - 2^(2/3)*3^(1/6)*(27 + sqrt(81 - 6*x))*x^(1/3))/(48*Pi*(9 + sqrt(81 - 6*x))^(1/3)*x^(2/3)).
This integral representation is unique as W(x) is the solution of the Hausdorff power moment problem for x on (0, 27/2). Using only the definition of a(n), W(x) can be proven to be positive. W(x) is singular at x = 0, with singularity x^(-2/3), and for x > 0 is monotonically decreasing to zero at x = 27/2. (End)

Definition clarified by Michael Albert, Oct 24 2008

A176129 Number A(n,k) of solid standard Young tableaux of shape [[n*k,n],[n]]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 6, 16, 0, 1, 12, 174, 192, 0, 1, 20, 690, 7020, 2816, 0, 1, 30, 1876, 52808, 325590, 46592, 0, 1, 42, 4140, 229680, 4558410, 16290708, 835584, 0, 1, 56, 7986, 738192, 31497284, 420421056, 854630476, 15876096, 0

Offset: 0

Alois P. Heinz, Jul 29 2012

In general, column k is (for k > 1) asymptotic to sqrt((k+2)*(k^2 - 20*k - 8 + sqrt(k*(k+8)^3)) / (8*k^3)) * ((k+2)^(k+2)/k^k)^n / (Pi*n). - Vaclav Kotesovec, Aug 31 2014

			Square array A(n,k) begins:
  1,      1,        1,         1,          1,           1, ...
  0,      2,        6,        12,         20,          30, ...
  0,     16,      174,       690,       1876,        4140, ...
  0,    192,     7020,     52808,     229680,      738192, ...
  0,   2816,   325590,   4558410,   31497284,   146955276, ...
  0,  46592, 16290708, 420421056, 4600393936, 31113230148, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau

Columns k=0-10 give: A000007, A006335, A214801, A215686, A246619, A246620, A246621, A246632, A246633, A246634, A246635.
Rows n=0-3 give: A000012, A002378, A215687, A215688.
Main diagonal gives: A215123.

b:= proc(x, y, z) option remember; `if`(z>y, b(x, z, y), `if`(z>x, 0,
      `if`({x, y, z}={0}, 1, `if`(x>y and x>z, b(x-1, y, z), 0)+
      `if`(y>0, b(x, y-1, z), 0)+ `if`(z>0, b(x, y, z-1), 0))))
    end:
A:= (n, k)-> b(n*k, n, n):
seq(seq(A(n, d-n), n=0..d), d=0..8);

b [x_, y_, z_] := b[x, y, z] = If[z > y, b[x, z, y], If[z > x, 0, If[Union[{x, y, z}] == {0}, 1, If[x > y && x > z, b[x-1, y, z], 0] + If[y > 0, b[x, y-1, z], 0] + If[z > 0, b[x, y, z-1], 0]]]]; a[n_, k_] := b[n*k, n, n]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 8}] // Flatten (* Jean-François Alcover, Dec 11 2013, translated from Maple *)

A214722 Number A(n,k) of solid standard Young tableaux of shape [[{n}^k],[n]]; square array A(n,k), n>=0, k>=1, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 16, 5, 1, 4, 91, 192, 14, 1, 5, 456, 5471, 2816, 42, 1, 6, 2145, 143164, 464836, 46592, 132, 1, 7, 9724, 3636776, 75965484, 48767805, 835584, 429, 1, 8, 43043, 91442364, 12753712037, 55824699632, 5900575762, 15876096, 1430

Offset: 0

Alois P. Heinz, Jul 26 2012

			Square array A(n,k) begins:
   1,     1,        1,           1,              1,                 1, ...
   1,     2,        3,           4,              5,                 6, ...
   2,    16,       91,         456,           2145,              9724, ...
   5,   192,     5471,      143164,        3636776,          91442364, ...
  14,  2816,   464836,    75965484,    12753712037,     2214110119572, ...
  42, 46592, 48767805, 55824699632, 70692556053053, 98002078234748974, ...

Alois P. Heinz, Antidiagonals n = 0..20, flattened
S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau

Columns k=1-4 give: A000108, A006335, A213978, A215220.
Rows n=0-3 give: A000012, A000027, A214824, A211505.
A(n,n) gives A258583.
Cf. A213932, A214637, A214631, A258586.

b:= proc(l) option remember; local m; m:= nops(l);
      `if`({map(x-> x[], l)[]}={0}, 1, add(add(`if`(l[i][j]>
      `if`(i=m or nops(l[i+1])
      `if`(nops(l[i])=j, 0, l[i][j+1]), b(subsop(i=subsop(
       j=l[i][j]-1, l[i]), l)), 0), j=1..nops(l[i])), i=1..m))
    end:
A:= (n, k)-> b([[n$k], [n]]):
seq(seq(A(n, 1+d-n), n=0..d), d=0..10);

b[l_List] := b[l] = With[{m = Length[l]}, If[Union[Flatten[l]] == {0}, 1, Sum[Sum[If[l[[i, j]] > If[i == m || Length[l[[i+1]]] < j, 0, l[[i+1, j]]] && l[[i, j]] > If[Length[l[[i]]] == j, 0, l[[i, j+1]]], b[ReplacePart[l, i -> ReplacePart[l[[i]], j -> l[[i, j]] - 1]]], 0], {j, 1, Length[l[[i]]]}], {i, 1, m}]] ]; a[n_, k_] := b[{Array[n&, k], {n}}]; Table[Table[a[n, 1+d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 17 2013, translated from Maple *)

A215204 Number A(n,k) of solid standard Young tableaux of cylindrical shape lambda X k, where lambda ranges over all partitions of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 4, 4, 5, 1, 1, 10, 26, 10, 7, 1, 1, 28, 276, 258, 26, 11, 1, 1, 84, 3740, 14318, 3346, 76, 15, 1, 1, 264, 58604, 1161678, 1214358, 54108, 232, 22, 1, 1, 858, 1010616, 118316062, 741215012, 150910592, 1054256, 764, 30

Offset: 0

Alois P. Heinz, Aug 05 2012

			Square array A(n,k) begins:
:  1,  1,     1,         1,            1,                1, ...
:  1,  1,     1,         1,            1,                1, ...
:  2,  2,     4,        10,           28,               84, ...
:  3,  4,    26,       276,         3740,            58604, ...
:  5, 10,   258,     14318,      1161678,        118316062, ...
:  7, 26,  3346,   1214358,    741215012,     620383261034, ...
: 11, 76, 54108, 150910592, 840790914296, 7137345113624878, ...

Alois P. Heinz, Antidiagonals n = 0..15, flattened
S. B. Ekhad and D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau

Columns k=0-5 give: A000041, A000085, A215266, A290202, A290214, A290274.
Rows n=0+1, 2-5 give: A000012, 2*A000108, 2*A005789 + A006335, 2*A005790 + 2*A213978 + A114714, 2*A005791 + 2*A215220 + 2*A213932 + A214638.
Main diagonal gives A290225.

b:= proc(l) option remember; local m; m:= nops(l);
      `if`({map(x-> x[], l)[]}minus{0}={}, 1, add(add(`if`(l[i][j]>
      `if`(i=m or nops(l[i+1])
      `if`(nops(l[i])=j, 0, l[i][j+1]), b(subsop(i=subsop(
       j=l[i][j]-1, l[i]), l)), 0), j=1..nops(l[i])), i=1..m))
    end:
g:= proc(n, i, k, l) `if`(n=0 or i=1, b(map(x-> [k$x], [l[], 1$n])),
       add(g(n-i*j, i-1, k, [l[], i$j]), j=0..n/i))
    end:
A:= (n, k)-> g(n, n, k, []):
seq(seq(A(n, d-n), n=0..d), d=0..10);

b[l_] := b[l] = With[{m = Length[l]}, If[Union[l // Flatten] ~Complement~ {0} == {}, 1, Sum[Sum[If[l[[i, j]] > If[i == m || Length[l[[i + 1]]] < j, 0, l[[i + 1, j]]] && l[[i, j]] > If[Length[l[[i]]] == j, 0, l[[i, j + 1]]], b[ReplacePart[l, i -> ReplacePart[l[[i]], j -> l[[i, j]] - 1]]], 0], {j, 1, Length[l[[i]]]}], {i, 1, m}]]];
g[n_, i_, k_, l_] := If[n == 0 || i == 1, b[Table[k, {#}]& /@ Join[l, Table[1, {n}]]], Sum[g[n - i*j, i - 1, k, Join[l, Table[i, {j}]]], {j, 0, n/i}]];
A[n_, k_] := g[n, n, k, {}];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Sep 24 2022, after Alois P. Heinz *)

A340591 Number A(n,k) of n*(k+1)-step k-dimensional nonnegative closed lattice walks starting at the origin and using steps that increment all components or decrement one component by 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 6, 16, 5, 1, 1, 24, 288, 192, 14, 1, 1, 120, 9216, 24444, 2816, 42, 1, 1, 720, 460800, 7303104, 2738592, 46592, 132, 1, 1, 5040, 33177600, 4234233600, 8204167296, 361998432, 835584, 429, 1, 1, 40320, 3251404800, 4223111040000, 59027412643200, 11332298092032, 53414223552, 15876096, 1430, 1

Offset: 0

Alois P. Heinz, Jan 12 2021

			Square array A(n,k) begins:
  1,  1,     1,         1,              1,                   1, ...
  1,  1,     2,         6,             24,                 120, ...
  1,  2,    16,       288,           9216,              460800, ...
  1,  5,   192,     24444,        7303104,          4234233600, ...
  1, 14,  2816,   2738592,     8204167296,      59027412643200, ...
  1, 42, 46592, 361998432, 11332298092032, 1052109889288796160, ...

Alois P. Heinz, Antidiagonals n = 0..24, flattened

Columns k=0-3 give: A000012, A000108, A006335, A340540.
Rows n=0-2 give: A000012, A000142, |A055546|.
Main diagonal gives A340590.
Cf. A335570.

b:= proc(n, l) option remember; `if`(n=0, 1, (k-> add(
     `if`(l[i]>0, b(n-1, sort(subsop(i=l[i]-1, l))), 0), i=1..k)+
     `if`(add(i, i=l)+k x+1, l)), 0))(nops(l)))
    end:
A:= (n, k)-> b(k*n+n, [0$k]):
seq(seq(A(n, d-n), n=0..d), d=0..10);

b[n_, l_] := b[n, l] = If[n == 0, 1, Function[k, Sum[
  If[l[[i]]>0, b[n-1, Sort[ReplacePart[l, i -> l[[i]]-1]]], 0], {i, 1, k}]+
  If[Sum[i, {i, l}] + k < n, b[n - 1, Map[#+1&, l]], 0]][Length[l]]];
A[n_, k_] := b[k*n + n, Table[0, {k}]];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Jan 26 2021, after Alois P. Heinz *)

A140136 Numerator coefficients for generators of lattice path enumeration square array A111910.

Original entry on oeis.org

1, 1, 1, 1, 7, 7, 1, 1, 20, 75, 75, 20, 1, 1, 42, 364, 1001, 1001, 364, 42, 1, 1, 75, 1212, 6720, 15288, 15288, 6720, 1212, 75, 1, 1, 121, 3223, 30723, 127908, 255816, 255816, 127908, 30723, 3223, 121, 1, 1, 182, 7371, 109538, 737737, 2510508

Offset: 0

Paul Barry, May 09 2008

			Triangle begins:
  1;
  1,  1;
  1,  7,    7,     1;
  1, 20,   75,    75,    20,     1;
  1, 42,  364,  1001,  1001,   364,   42,    1;
  1, 75, 1212,  6720, 15288, 15288, 6720, 1212, 75, 1;
  ...

Michael De Vlieger, Table of n, a(n) for n = 0..10100 (rows 0..100, flattened)
G. Kreweras, Sur une classe de problèmes de dénombrement liés au treillis des partitions des entiers, Cahiers du B.U.R.O. 6 (1965), 9-107; see p. 93.
G. Kreweras and H. Niederhausen, Solution of an enumerative problem connected with lattice paths, European J. Combin. 2 (1981), 55-60; see p. 60.
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 8.

Row sums are A006335.

Cf. A111910.

T[n_, k_] := ((k + n - 1)! (2 (k + n) - 3)! HypergeometricPFQ[{2 - 3 k, 1/2 - n, 1 - n, -n}, {1 - k - n, 3/2 - k - n, 2 - k - n}, 1])/(k! (2 k - 1)! n! (2 n - 1)!);
Join[{{1}}, Table[T[n, k], {k, 2, 8}, {n, 1, 2 k - 2}]] // Flatten (* Peter Luschny, Sep 04 2019 *)

(Sum_{k=0..n} T(n,k) * x^k) / (1-x)^(3*n+1) generates row n of A111910.

Triangle T(q,n), where T(n,q) = Sum_{j = 0..n} (-1)^j*C(3*q+1,j)*K(n-j,q) with K(p,q) = A111910(p,q).

A214631 Number A(n,k) of solid standard Young tableaux of shape [[(n)^(k+1)],[n]^k]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 6, 16, 1, 1, 20, 936, 192, 1, 1, 70, 85800, 379366, 2816, 1, 1, 252, 9962680, 1825221320, 249664758, 46592, 1, 1, 924, 1340103744, 14336196893200, 89261675900020, 221005209058, 835584, 1

Offset: 0

Alois P. Heinz, Jul 26 2012

			Square array A(n,k) begins:
  1,    1,         1,              1,                    1, ...
  1,    2,         6,             20,                   70, ...
  1,   16,       936,          85800,              9962680, ...
  1,  192,    379366,     1825221320,       14336196893200, ...
  1, 2816, 249664758, 89261675900020, 70351928759681296000, ...

Alois P. Heinz, Antidiagonals n = 0..12
S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau

Columns k=0-2 give: A000012, A006335, A214638.
Rows n=0-1 give: A000012, A000984.
Cf. A213932, A213978, A214637, A214722.

b:= proc(l) option remember; local m; m:= nops(l);
      `if`({map(x-> x[], l)[]}={0}, 1, add(add(`if`(l[i][j]>
      `if`(i=m or nops(l[i+1])
      `if`(nops(l[i])=j, 0, l[i][j+1]), b(subsop(i=subsop(
       j=l[i][j]-1, l[i]), l)), 0), j=1..nops(l[i])), i=1..m))
    end:
A:= (n, k)-> b([[n$(k+1)], [n]$k]):
seq(seq(A(n, d-n), n=0..d), d=0..8);

b[l_] := b[l] = With[{m = Length[l]}, If[Union[Flatten[l]] == {0}, 1, Sum[Sum[If[l[[i, j]] > If[i == m || Length[l[[i+1]] ] < j, 0, l[[i+1, j]] ] && l[[i, j]] > If[Length[l[[i]] ] == j, 0, l[[i, j+1]] ], b[ReplacePart[l, i -> ReplacePart[l[[i]], j -> l[[i, j]]-1]]], 0], {j, 1, Length[l[[i]] ]}], {i, 1, m}]]]; a[n_, k_] := b[{Array[n&, k+1], Sequence @@ Array[{n}&, k]}]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 8}] // Flatten (* Jean-François Alcover, Dec 18 2013, translated from Maple *)

A098272 a(n) = 2^(2n+1) * binomial(3n,n)/(2n+1).

Original entry on oeis.org

2, 8, 96, 1536, 28160, 559104, 11698176, 254017536, 5670567936, 129328742400, 3000426823680, 70587116421120, 1679973370822656, 40376795886780416, 978590323955466240, 23890230876435382272, 586939535850605641728

Offset: 0

Ralf Stephan, Sep 02 2004

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..100
M. Bousquet-Mélou, Walks in the quarter plane: Kreweras' algebraic model, arXiv:math/0401067 [math.CO], 2004-2006.

Cf. A001764, A006335.

[2^(2*n+1)*Binomial(3*n, n)/(2*n+1): n in [0..20]]; // Vincenzo Librandi, Oct 03 2011

Table[2^(2n+1) Binomial[3n,n]/(2n+1),{n,0,20}] (* Harvey P. Dale, Oct 02 2011 *)

PARI
```
a(n)=2^(2*n+1)*binomial(3*n,n)/(2*n+1)
```

a(n)=polcoeff(serreverse(Ser(x/(2+x^3))),3*n+1)

G.f. satisfies A(x) = Sum_{n>=0} a(n)*x^(3n+1) = x(2 + A(x)^3).
a(n) = 2n * A006335(n) = 2^(2n+1) * A001764(n).
G.f.: (2 sin(1/3*arcsin(3*sqrt(3)*sqrt(x))))/(sqrt(3)*sqrt(x)). - Harvey P. Dale, Oct 02 2011
E.g.f.: 2*2F2(1/3,2/3; 1,3/2; 27*x). - Ilya Gutkovskiy, Jan 25 2017

A340540 Number of walks of length 4n in the first octant using steps (1,1,1), (-1,0,0), (0,-1,0), and (0,0,-1) that start and end at the origin.

Original entry on oeis.org

1, 6, 288, 24444, 2738592, 361998432, 53414223552, 8525232846072, 1443209364298944, 255769050813120576, 47020653859202576640, 8907614785269428079168, 1730208409741026141405696, 343266632435192859791576064, 69350551439109880798294334208

Offset: 0

Daniel Carter, Jan 10 2021

There are no such walks with length that is not a multiple of 4.
a(n) is also the number of arrangements of n copies each of "a", "b", "c", and "d" such that no prefix has more b's, c's, or d's than a's.
The analogous problem in dimensions 1 and 2 are given respectively by A000108 (the Catalan numbers) and A006335.
No closed form is known. In fact, it is not known whether this sequence is D-finite (see Bacher et al.).

Alois P. Heinz, Table of n, a(n) for n = 0..150
Axel Bacher, Manuel Kauers, and Rika Yatchak, Continued Classification of 3D Lattice Walks in the Positive Octant, arXiv:1511.05763 [math.CO], 2015.

Cf. A000108, A006335, A149424.

Column k=3 of A340591.

b:= proc(n, l) option remember; `if`(n=0, 1, `if`(add(i, i=l)+3 x+1, l)), 0) +add(`if`(l[i]>0,
      b(n-1, sort(subsop(i=l[i]-1, l))), 0), i=1..3))
    end:
a:= n-> b(4*n, [0$3]):
seq(a(n), n=0..15);  # Alois P. Heinz, Jan 12 2021

b[n_, l_] := b[n, l] = If[n == 0, 1, If[Total[l] + 3 < n,
  b[n-1, l+1]], 0] + Sum[If[l[[i]] > 0,
  b[n-1, Sort[ReplacePart[l, i -> l[[i]]-1]]], 0], {i, 1, 3}] /. Null -> 0;
a[n_] := b[4n, {0, 0, 0}];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Jul 10 2021, after Alois P. Heinz *)

import itertools as it
i = 0
while 1:
    counts = {(a,b,c):0 for a,b,c in it.product(range(i+1), repeat=3)}
    counts[0,0,0] = 1
    for _ in range(4*i):
        update = {(a,b,c):0 for a,b,c in it.product(range(i+1), repeat=3)}
        for x,y,z in counts:
            if counts[x,y,z] != 0:
                for coord in [(x+1,y+1,z+1), (x-1,y,z), (x,y-1,z), (x,y,z-1)]:
                    if coord in update:
                        update[coord] += counts[x,y,z]
        counts = update
    print(i, counts[0,0,0])
    i += 1

cp's OEIS Frontend

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

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A000309 Number of rooted planar bridgeless cubic maps with 2n nodes.

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A176129 Number A(n,k) of solid standard Young tableaux of shape [[n*k,n],[n]]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A214722 Number A(n,k) of solid standard Young tableaux of shape [[{n}^k],[n]]; square array A(n,k), n>=0, k>=1, read by antidiagonals.

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A215204 Number A(n,k) of solid standard Young tableaux of cylindrical shape lambda X k, where lambda ranges over all partitions of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A340591 Number A(n,k) of n*(k+1)-step k-dimensional nonnegative closed lattice walks starting at the origin and using steps that increment all components or decrement one component by 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A140136 Numerator coefficients for generators of lattice path enumeration square array A111910.

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A000139 a(n) = 2(3n)! / ((2n+1)!(n+1)!).