A210664 -id:A210664 - cp's OEIS frontend

A004305 Simple triangulations of a disk: column 4 of square array in A210664.

3, 22, 85, 254, 644, 1448, 2967, 5645, 10109, 17214, 28093, 44212, 67430, 100064, 144959, 205563, 286007, 391190, 526869, 699754, 917608, 1189352, 1525175, 1936649, 2436849, 3040478, 3763997, 4625760, 5646154, 6847744, 8255423, 9896567, 11801195, 14002134, 16535189, 19439318, 22756812, 26533480, 30818839

Offset: 0

N. J. A. Sloane

nonn

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

Andrew Howroyd, Table of n, a(n) for n = 0..1000
P. N. Rathie, A census of simple planar triangulations, J. Combin. Theory, B 16 (1974), 134-138.

Cf. A210664.

a(n) = (n^6 + 15*n^5 + 103*n^4 + 405*n^3 + 976*n^2 + 1236*n + 432)/144.

Entry revised by N. J. A. Sloane, Mar 27 2012

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

Original entry on oeis.org

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)

A341856 Array read by antidiagonals: T(n,k) is the number of rooted strong triangulations of a disk with n interior nodes and 3+k nodes on the boundary.

Original entry on oeis.org

1, 0, 1, 0, 1, 3, 0, 1, 6, 13, 0, 1, 10, 36, 68, 0, 1, 15, 80, 228, 399, 0, 1, 21, 155, 610, 1518, 2530, 0, 1, 28, 273, 1410, 4625, 10530, 16965, 0, 1, 36, 448, 2933, 12165, 35322, 75516, 118668, 0, 1, 45, 696, 5628, 28707, 102548, 272800, 556512, 857956

Offset: 0

Andrew Howroyd, Feb 23 2021

A strong triangulation is one in which no interior edge joins two nodes on the boundary. Except for the single triangle which is enumerated by T(0,0) these are the 3-connected triangulations.

			Array begins:
=======================================================
n\k |    0     1     2      3      4      5       6
----+--------------------------------------------------
  0 |    1     0     0      0      0      0       0 ...
  1 |    1     1     1      1      1      1       1 ...
  2 |    3     6    10     15     21     28      36 ...
  3 |   13    36    80    155    273    448     696 ...
  4 |   68   228   610   1410   2933   5628   10128 ...
  5 |  399  1518  4625  12165  28707  62230  125928 ...
  6 | 2530 10530 35322 102548 267162 638624 1422204 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325
William T. Tutte, A census of planar triangulations, Canad. J. Math. 14 (1962), 21-38.

Columns k=0..3 are A000260, A242136, A341917, A341918.
Antidiagonal sums give A341919.
Cf. A146305 (not necessarily strong triangulations), A210664, A341923, A342053.

T(n,m)=if(m==0, 2*(4*n+1)!/((3*n+2)!*(n+1)!), (3*(m+2)!*(m-1)!/(3*n+3*m+3)!)*sum(j=0, min(m,n-1), (4*n+3*m-j+1)!*(m+j+2)*(m-3*j)/(j!*(j+1)!*(m-j)!*(m-j+2)!*(n-j-1)!)))

T(n,0) = A000260(n) = 2*(4*n+1)!/((3*n+2)!*(n+1)!).

T(n,m) = (3*(m+2)!*(m-1)!/(3*n+3*m+3)!) * Sum_{j=0..min(m,n-1)} (4*n+3*m-j+1)!*(m+j+2)*(m-3*j)/(j!*(j+1)!*(m-j)!*(m-j+2)!*(n-j-1)!) for m > 0.

A000256 Number of simple triangulations of the plane with n nodes.

Original entry on oeis.org

1, 1, 0, 1, 3, 12, 52, 241, 1173, 5929, 30880, 164796, 897380, 4970296, 27930828, 158935761, 914325657, 5310702819, 31110146416, 183634501753, 1091371140915, 6526333259312, 39246152584304, 237214507388796, 1440503185260748

Offset: 3

N. J. A. Sloane

A triangulation is simple if it contains no separating 3-cycle. The triangulations are rooted with three fixed exterior nodes. - Andrew Howroyd, Feb 24 2021

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).
W. T. Tutte, The enumerative theory of planar maps, pp. 437-448 of J. N. Srivastava, ed., A Survey of Combinatorial Theory, North-Holland, 1973.

T. D. Noe, Table of n, a(n) for n=3..200
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.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, Une méthode pour obtenir la fonction génératrice d'une série. FPSAC 1993, Florence. Formal Power Series and Algebraic Combinatorics; arXiv:0912.0072 [math.NT], 2009.
P. N. Rathie, A census of simple planar triangulations, J. Combin. Theory, B 16 (1974), 134-138.
William T. Tutte, A census of planar triangulations, Canad. J. Math. 14 (1962), 21-38.
William T. Tutte, A Census of Planar Maps, Canad. J. Math. 15 (1963), 249-271.

First row of array in A210664.

R := RootOf(x-t*(t-1)^2, t); ogf := series( (2*R^3+2*R^2-2*R-1)/((R-1)*(R+1)^2), x=0, 20); # Mark van Hoeij, Nov 08 2011

r = Root[x - t*(t - 1)^2, t, 1] ; CoefficientList[ Series[(2*r^3 + 2*r^2 - 2*r - 1)/((r - 1)*(r + 1)^2), {x, 0, 24}], x] (* Jean-François Alcover, Mar 14 2012, after Maple *)

A000260_ser(N) = {
  my(v = vector(N, n, binomial(4*n+2, n+1)/((2*n+1)*(3*n+2))));
  Ser(concat(1,v));
};
A000256_seq(N) = {
  my(g = A000260_ser(N)); Vec(subst(2 - 1/g, 'x, serreverse(x*g^2)));
};
A000256_seq(24)
\\ test: y = Ser(A000256_seq(200)); 0 == x*(x+4)^2*y^3 - x*(6*x^2+51*x+76)*y^2 + (12*x^3+108*x^2+115*x-1)*y - (8*x^3+76*x^2+54*x-1)
\\ Gheorghe Coserea, Jul 31 2017

seq(n)={my(g=1+serreverse(x/(1+x)^4 + O(x*x^n) )); Vec(2 - sqrt(serreverse( x*(2-g)^2*g^4)/x ))} \\ Andrew Howroyd, Feb 23 2021

a(n) = (1/4)*(7*binomial(3*n-9, n-4)-(8*n^2-43*n+57)*a(n-1)) / (8*n^2-51*n+81), n>4. - Vladeta Jovovic, Aug 19 2004
(1/4 + 7/8*n - 9/8*n^3)*a(n) + (-5/4 + 2/3*n + 59/12*n^2 - 13/3*n^3)*a(n+1) + (-1 - 2/3*n + n^2 + 2/3*n^3)*a(n+2). - Simon Plouffe, Feb 09 2012
a(n) ~ 3^(3*n-6+1/2)/(2^(2*n+3)*sqrt(Pi)*n^(5/2)). - Vaclav Kotesovec, Aug 13 2013
From Gheorghe Coserea, Jul 31 2017: (Start)
G.f. y(x) satisfies (with offset 0):
y(x*g^2) = 2 - 1/g, where g=A000260(x). (eqn 2.6 in Tutte's paper)
0 = x*(x+4)^2*y^3 - x*(6*x^2+51*x+76)*y^2 + (12*x^3+108*x^2+115*x-1)*y - (8*x^3+76*x^2+54*x-1).
0 = x*(27*x-4)*deriv(y,x) + x*(7*x+28)*y^2 - 2*(14*x^2+45*x+1)*y + 2*(14*x^2+34*x+1).
(End)

More terms from Vladeta Jovovic, Aug 19 2004

A341920 Number of simple strong triangulations of a fixed pentagon with n interior nodes.

Original entry on oeis.org

1, 5, 20, 85, 385, 1836, 9120, 46805, 246675, 1328925, 7292688, 40652780, 229698220, 1313175880, 7585060680, 44212723797, 259808593975, 1537840085775, 9162371040900, 54913015836525, 330888232445505, 2003659057873200, 12187811305300800, 74444226847892460, 456458831442283860

Offset: 1

Andrew Howroyd, Feb 24 2021

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..500

Row m = 2 of A210664.

Cf. A000256, A341917.

PARI
```
R(25,2) \\ R defined in A210664.
```

From Peter Bala, Feb 06 2022: (Start)
Conjectures:
a(n) = 5*(5*n-3)*binomial(3*n,n)/((n+1)*(2*n+1)*(2*n+3)).
a(n+1) = 3*(3*n+1)*(3*n+2)*(5*n+2)/(2*(n+2)*(2*n+5)*(5*n-3))*a(n). (End)

A341921 Number of simple strong triangulations of a fixed hexagon with n interior nodes.

Original entry on oeis.org

1, 9, 50, 254, 1287, 6630, 34846, 186714, 1018095, 5638035, 31651992, 179850344, 1032901028, 5988670002, 35017696806, 206326044386, 1224068679855, 7307391855855, 43871253636210, 264755875041630, 1605357997312335, 9776732655285960, 59781214452500424, 366905787075963864

Offset: 1

Andrew Howroyd, Feb 24 2021

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..500

Row m=3 of A210664.

Cf. A341918.

PARI
```
R(25,3) \\ R defined in A210664.
```

A341922 Number of simple 3-connected triangulations of a disk with n nodes.

Original entry on oeis.org

1, 1, 4, 15, 64, 293, 1414, 7102, 36809, 195676, 1062176, 5867676, 32901124, 186866589, 1073261418, 6225028476, 36420849235, 214744648169, 1275005642668, 7617704400227, 45772485843616, 276460659973100, 1677717692649544, 10225697003966360, 62575895785795084

Offset: 4

Andrew Howroyd, Feb 24 2021

nonn

A triangulation is simple if it contains no separating 3-cycle.

			The a(5) = 1 triangulation is:
      o-------o
      | \   / |
      |   o   |
      | /   \ |
      o-------o

Andrew Howroyd, Table of n, a(n) for n = 4..200

Antidiagonal sums of A210664.

\\ R(n,m) defined in A210664.
seq(n)={my(v=R(n,0)); for(m=1, n-1, v+=concat(vector(m), R(n-m,m))); v}

cp's OEIS Frontend

A004305 Simple triangulations of a disk: column 4 of square array in A210664.

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

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A341856 Array read by antidiagonals: T(n,k) is the number of rooted strong triangulations of a disk with n interior nodes and 3+k nodes on the boundary.

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A000256 Number of simple triangulations of the plane with n nodes.

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A341920 Number of simple strong triangulations of a fixed pentagon with n interior nodes.

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A341921 Number of simple strong triangulations of a fixed hexagon with n interior nodes.

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A341922 Number of simple 3-connected triangulations of a disk with n nodes.

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