A262586 -id:A262586 - cp's OEIS frontend

A001683 Number of one-sided triangulations of the disk; or flexagons of order n; or unlabeled plane trivalent trees (n-2 internal vertices, all of degree 3 and hence n leaves).

1, 1, 1, 1, 4, 6, 19, 49, 150, 442, 1424, 4522, 14924, 49536, 167367, 570285, 1965058, 6823410, 23884366, 84155478, 298377508, 1063750740, 3811803164, 13722384546, 49611801980, 180072089896, 655977266884, 2397708652276, 8791599732140, 32330394085528

Offset: 2

N. J. A. Sloane

a(n) is the number of triangulations of an n-gon (equivalently, the number of vertices of the (n - 3)-dimensional associahedron) modulo the cyclic action [Bowman and Regev]. - N. J. A. Sloane, Dec 29 2012
a(n) is also the number of non-isomorphic cluster-tilted algebras of type A_(n-3), for n greater than or equal to 5. Equivalently it is the number of non-isomorphic quivers in the mutation class of any quiver with underlying graph A_(n-3) for n greater than or equal to 5. - Hermund A. Torkildsen (hermunda(AT)math.ntnu.no), Aug 06 2008
Number of oriented polyominoes composed of n-2 triangular cells of the hyperbolic regular tiling with Schläfli symbol {3,oo}. A stereographic projection of this tiling on the Poincaré disk can be obtained via the Christensson link. For oriented polyominoes, chiral pairs are counted as two. - Robert A. Russell, Jan 20 2024

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=2..200
Marc J. Beauchamp, On Extremal Punctured Spheres, Dissertation, University of Pittsburgh, 2017.
F. R. Bernhart & N. J. A. Sloane, Correspondence, 1977
Douglas Bowman and Alon Regev, Counting symmetry classes of dissections of a convex regular polygon, arXiv preprint arXiv:1209.6270 [math.CO], 2012. See Th. 29(2).
William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768.
W. G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768. [Annotated scanned copy]
P. J. Cameron, Some treelike objects, Quart. J. Math. Oxford, 38 (1987), 155-183. See p. 163, line 4, but note that the formula given there has many typos (see the correct version given here).
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Malin Christensson, Make hyperbolic tilings of images, web page, 2019.
O. Devillers, Vertex removal in two-dimensional Delauney triangulation: Speed-up by low degrees optimization, Comp. Geom. 44 (2011) 169.
Petr Gregor, Sven Jäger, Torsten Mütze, Joe Sawada, Kaja Wille, Gray codes and symmetric chains, arXiv:1802.06021 [math.CO], 2018.
F. Harary, E. M. Palmer, R. C. Read, On the cell-growth problem for arbitrary polygons, computer printout, circa 1974
F. Harary, E. M. Palmer and R. C. Read, On the cell-growth problem for arbitrary polygons, Discr. Math. 11 (1975), 371-389.
E. Krasko, A. Omelchenko, Brown's Theorem and its Application for Enumeration of Dissections and Planar Trees, The Electronic Journal of Combinatorics, 22 (2015), #P1.17.
C. O. Oakley and R. J. Wisner, Flexagons, The American Mathematical Monthly, Vol. 64, No. 3 (Mar., 1957), pp. 143-154
R. C. Read, On general dissections of a polygon, Preprint (1974)
Hermund A. Torkildsen, Counting cluster-tilted algebras of type A_n, International Electronic Journal of Algebra, 4, 2008, 149-158. [From Hermund A. Torkildsen (hermunda(AT)math.ntnu.no), Aug 06 2008]
Hermund A. Torkildsen, Colored quivers of type A and the cell-growth problem, J. Algebra and Applications, 12 (2013), #1250133. - From _N. J. A. Sloane_, Jan 22 2013

Column k=3 of A295224.
Cf. A007282, A057162.
A row or column of the array in A262586.
Polyominoes: A000207 (unoriented), A369314 (chiral), A208355(n-1) (achiral), A005034 {4,oo}, A007173 {3,3,oo}.

C := n->binomial(2*n,n)/(n+1); c := x->if whattype(x) = integer then C(x) else 0; fi; A001683 := n->C(n-2)/n + c(n/2-1)/2+(2/3)*c(n/3-1);

p=3; Table[Binomial[(p-1)n, n]/(((p-2)n+1)((p-2)n+2)) +If[OddQ[n], 0, Binomial[(p-1)n/2, n/2]/((p-2)n+2)]+Plus @@ Map[EulerPhi[ # ]Binomial[((p-1)n+1)/#, (n-1)/# ]/((p-1)n+1)&, Complement[Divisors[GCD[p, n-1]], {1}]], {n, 0, 20}] (* Robert A. Russell, Dec 11 2004 *)
Rest[Rest[CoefficientList[Series[(6 + (1 - 4 x)^(3/2) + 6 x - 3(1 - 4 x^2)^(1/2) - 4 (1 - 4 x^3)^(1/2))/12, {x, 0, 33}], x]]] (* Vincenzo Librandi, Nov 25 2015 *)

Cat(n)=if(n==floor(n),return(binomial(2*n,n)/(n+1)));0
for(n=2,100,print1(Cat(n-2)/n+Cat(n/2-1)/2+(2/3)*Cat(n/3-1),", ")) \\ Derek Orr, Feb 26 2017

a(n) = C(n-2)/n + C(n/2-1)/2 + (2/3)*C(n/3-1), where C(n) = Catalan(n) (A000108) and terms are omitted if their subscripts are not integers.
G.f.: (6 + (1 - 4*x)^(3/2) + 6*x - 3*(1 - 4*x^2)^(1/2) - 4*(1 - 4*x^3)^(1/2))/12. - David Callan, Aug 01 2004
a(n) ~ 2^(2*n-4) / (sqrt(Pi) * n^(5/2)). - Vaclav Kotesovec, Mar 13 2016
a(n+2) = A000207(n) + A369314(n) = 2*A000207(n) - A208355(n-1) = 2*A369314(n) + A208355(n-1). - Robert A. Russell, Jan 19 2024
G.f.: z^2 * (4*G(z) - G(z)^2 + 3*G(z^2) + 4*z*G(z^3)) / 6, where G(z) = 1 + z*G(z)^2 is the g.f. for A000108. - Robert A. Russell, Apr 06 2024

A002709 Triangulations of the disk G_{n,0}.

Original entry on oeis.org

1, 1, 1, 5, 24, 133, 846, 5661, 39556, 286000, 2123329, 16112057, 124512556, 977227830, 7772368380, 62535450861, 508271324688, 4168218286276, 34455941596060, 286864341314320, 2403705165816240, 20258850167232165, 171652324167433710, 1461462393790971585, 12498416291503945764

Offset: 0

N. J. A. Sloane

nonn

Apparently, also the number of regular flexagons of order 3(n+1) (see Oakley-Wisner link pp. 149-151). - Michel Marcus, Jun 23 2013

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).

Andrew Howroyd, Table of n, a(n) for n = 0..500
William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768.
William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768. [Annotated scanned copy]
C. O. Oakley and R. J. Wisner, Flexagons, The American Mathematical Monthly, Vol. 64, No. 3 (Mar., 1957), pp. 143-154.

Column k=0 of A262586.

a(n) = {if (n % 3 == 0, k = n/3; return (binomial(12*k-1,3*k-1)/((6*k-1)*(12*k-1)));); if (n % 3 == 1, k = (n-1)/3; return (binomial(12*k+3, 3*k)/(3*(4*k+1)*(6*k+1))+2*binomial(4*k,k)/(3*(3*k+1)));); if (n % 3 == 2, k = (n-2)/3; return (binomial(12*k+7,3*k+1)/(3*(2*k+1)*(12*k+7))+4*binomial(4*k+1,k)/(3*(3*k+2))););} \\ (number of regular flexagons of order 3*n) Michel Marcus, Jun 15 2013

Extended by Max Alekseyev, Mar 30 2009

a(22) onwards from Andrew Howroyd, Nov 23 2024

A169808 Array T(n,k) read by antidiagonals: T(n,k) is the number of [n,k]-triangulations in the plane, n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 4, 5, 4, 4, 11, 14, 18, 16, 12, 28, 53, 69, 88, 78, 27, 91, 178, 295, 396, 489, 457, 82, 291, 685, 1196, 1867, 2503, 3071, 2938, 228, 1004, 2548, 5051, 8385, 12560, 16905, 20667, 20118, 733, 3471, 9876, 21018, 38078, 60736, 89038, 119571, 146381, 144113

Offset: 0

N. J. A. Sloane, May 25 2010

"A closed bounded region in the plane divided into triangular regions with k+3 vertices on the boundary and n internal vertices is said to be a triangular map of type [n,k]." It is a [n,k]-triangulation if there are no multiple edges.
T(n,k) is the number of floor plan arrangements represented by 3-connected trivalent maps with n internal rooms and k+3 rooms adjacent to the outside.
"... may be evaluated from the results given by Brown."
The initial terms of this sequence can also be computed using the tool "plantri", in particular the command "./plantri -u -v -P -c2m2 [n]" will compute values for a diagonal. The '-c2' and '-m2' options indicate graphs must be biconnected and with minimum vertex degree 2. - Andrew Howroyd, Feb 22 2021

			Array begins:
============================================================
n\k |    0     1      2      3       4        5        6
----+-------------------------------------------------------
  0 |    1     1      1      3       4       12       27 ...
  1 |    1     2      4     11      28       91      291 ...
  2 |    1     5     14     53     178      685     2548 ...
  3 |    4    18     69    295    1196     5051    21018 ...
  4 |   16    88    396   1867    8385    38078   169918 ...
  5 |   78   489   2503  12560   60736   290595  1367374 ...
  6 |  457  3071  16905  89038  451613  2251035 11025626 ...
  7 | 2938 20667 119571 652198 3429943 17658448 89328186 ...
  ...

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.

Andrew Howroyd, Table of n, a(n) for n = 0..1325
G. Brinkmann and B. McKay, Plantri (program for generation of certain types of planar graph)
William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768.
William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768. [Annotated scanned copy].
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)
Andrew Howroyd, PARI Program

Columns k=0..3 are A002713, A005500, A005501, A005502.
Rows n=0..2 are A000207, A005503, A005504.
Antidiagonal sums give A005027.
Cf. A146305 (rooted), A169809 (achiral), A262586 (oriented).

\\ See link for script file.
A169808Array(6) \\ Andrew Howroyd, Feb 22 2021

T(n,k) = (A262586(n,k) + A169809(n,k)) / 2. - Andrew Howroyd, Feb 22 2021

Edited by Andrew Howroyd, Feb 22 2021

a(29) corrected and terms a(36) and beyond from Andrew Howroyd, Feb 22 2021

A169809 Array T(n,k) read by antidiagonals: T(n,k) is the number of [n,k]-triangulations in the plane that have reflection symmetry, n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 3, 4, 3, 2, 6, 7, 10, 8, 5, 8, 18, 19, 29, 23, 5, 18, 26, 52, 57, 86, 68, 14, 23, 68, 82, 166, 176, 266, 215, 14, 56, 91, 220, 270, 524, 557, 844, 680, 42, 70, 248, 321, 769, 890, 1722, 1806, 2742, 2226, 42, 180, 318, 872, 1151, 2568, 2986, 5664, 5954, 9032, 7327

Offset: 0

N. J. A. Sloane, May 25 2010

"A closed bounded region in the plane divided into triangular regions with k+3 vertices on the boundary and n internal vertices is said to be a triangular map of type [n,k]." It is a [n,k]-triangulation if there are no multiple edges.

"... may be evaluated from the results given by Brown."

			Array begins:
====================================================
n\k |   0   1    2    3     4     5     6      7
----+-----------------------------------------------
  0 |   1   1    1    2     2     5     5     14 ...
  1 |   1   2    3    6     8    18    23     56 ...
  2 |   1   4    7   18    26    68    91    248 ...
  3 |   3  10   19   52    82   220   321    872 ...
  4 |   8  29   57  166   270   769  1151   3296 ...
  5 |  23  86  176  524   890  2568  4020  11558 ...
  6 |  68 266  557 1722  2986  8902 14197  42026 ...
  7 | 215 844 1806 5664 10076 30362 49762 148208 ...
  ...

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.

Andrew Howroyd, Table of n, a(n) for n = 0..1325
William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768.
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)

Columns k=0..3 are A002712, A005505, A005506, A005507.
Rows n=0..2 are A208355, A005508, A005509.
Antidiagonal sums give A005028.
Cf. A146305 (rooted), A169808 (unrooted), A262586 (oriented).

\\ See link in A169808 for script.
A169809Array(7) \\ Andrew Howroyd, Feb 22 2021

Edited and terms a(36) and beyond from Andrew Howroyd, Feb 22 2021

A146305 Array T(n,m) = 2(2m+3)!(4n+2m+1)!/(m!(m+2)!n!*(3n+2m+3)!) read by antidiagonals.

Original entry on oeis.org

1, 1, 2, 3, 5, 5, 13, 20, 21, 14, 68, 100, 105, 84, 42, 399, 570, 595, 504, 330, 132, 2530, 3542, 3675, 3192, 2310, 1287, 429, 16965, 23400, 24150, 21252, 16170, 10296, 5005, 1430, 118668, 161820, 166257, 147420, 115500, 78936, 45045, 19448, 4862, 857956

Offset: 0

R. J. Mathar, Oct 29 2008

T(n,m) is the number of rooted nonseparable (2-connected) triangulations of the disk with n internal nodes and 3 + m nodes on the external face. The triangulation has 2*n + m + 1 triangles and 3*(n+1) + 2*m edges. - Andrew Howroyd, Feb 21 2021

			The array starts at row n=0 and column m=0 as
.....1......2.......5......14.......42.......132
.....1......5......21......84......330......1287
.....3.....20.....105.....504.....2310.....10296
....13....100.....595....3192....16170.....78936
....68....570....3675...21252...115500....602316
...399...3542...24150..147420...844074...4628052
..2530..23400..166257.1057224..6301680..35939904
.16965.161820.1186680.7791168.47948670.282285432

Andrew Howroyd, Table of n, a(n) for n = 0..1325
Alin Bostan, Frédéric Chyzak, and Vincent Pilaud, Refined product formulas for Tamari intervals, arXiv:2303.10986 [math.CO], 2023.
William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768.

Columns m=0..3 are A000260, A197271(n+1), A341853, A341854.
Rows n=0..2 are A000108(n+1), A002054(n+1) and A000917.
Antidiagonal sums are A000260(n+1).
Cf. A169808 (unrooted), A169809 (achiral), A262586 (oriented).

A146305 := proc(n,m)
    2*(2*m+3)!*(4*n+2*m+1)!/m!/(m+2)!/n!/(3*n+2*m+3)! ;
end proc:
for d from 0 to 13 do
    for m from 0 to d do
        printf("%d,", A146305(d-m,m)) ;
    end do:
end do:

T[n_, m_] := 2*(2*m+3)!*(4*n+2*m+1)!/m!/(m+2)!/n!/(3*n+2*m+3)!; Table[T[n-m, m], {n, 0, 13}, {m, 0, n}] // Flatten (* Jean-François Alcover, Jan 06 2014, after Maple *)

T(n,m)={2*(2*m+3)!*(4*n+2*m+1)!/(m!*(m+2)!*n!*(3*n+2*m+3)!)} \\ Andrew Howroyd, Feb 21 2021

A341923 Array read by antidiagonals: T(n,k) is the number of 3-connected triangulations of a disk up to orientation-preserving isomorphisms with n interior nodes and k nodes on the boundary, n >= 1, k >= 3.

Original entry on oeis.org

1, 1, 1, 1, 2, 5, 1, 2, 10, 24, 1, 3, 16, 60, 133, 1, 3, 28, 122, 386, 846, 1, 4, 39, 242, 925, 2652, 5661, 1, 4, 58, 419, 2039, 7066, 18914, 39556, 1, 5, 78, 711, 4101, 17138, 54560, 139264, 286000, 1, 5, 106, 1128, 7801, 38166, 142802, 426462, 1048947, 2123329

Offset: 1

Andrew Howroyd, Feb 26 2021

The initial terms of this sequence can also be computed using the tool "plantri", in particular the command "./plantri -u -v -o -P [n]" will compute values for a diagonal.

			Array begins:
=====================================================
n\k |     3      4      5       6       7       8
----+------------------------------------------------
  1 |     1      1      1       1       1       1 ...
  2 |     1      2      2       3       3       4 ...
  3 |     5     10     16      28      39      58 ...
  4 |    24     60    122     242     419     711 ...
  5 |   133    386    925    2039    4101    7801 ...
  6 |   846   2652   7066   17138   38166   79908 ...
  7 |  5661  18914  54560  142802  345099  782210 ...
  8 | 39556 139264 426462 1188412 3067938 7433635 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275
G. Brinkmann and B. McKay, Plantri (program for generation of certain types of planar graph)

Columns k=3..6 are A002709, A341924, A341925, A341926.
Antidiagonal sums are A342052.
Cf. A262586 (2-connected), A341856 (rooted), A342053 (unrooted).

A341923Array(8,6) \\ See links in A342053 for program file.

A210696 Triangulations of the disk, G_{1,n}.

Original entry on oeis.org

1, 2, 5, 16, 48, 164, 559, 1952, 6872, 24520, 88006, 318444, 1158944, 4241688, 15598973, 57620596, 213680472, 795270644, 2969483214, 11121038100, 41763779054, 157235683780, 593355907790, 2243975358216, 8503404201874, 32283434698908, 122779218918272, 467713035691608

Offset: 0

R. J. Mathar, Mar 30 2012

nonn

This corrects a typographical error in A005497(6).

Andrew Howroyd, Table of n, a(n) for n = 0..500
Jean-François Alcover, Mathematica program
William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768.
William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768. [Annotated scanned copy]

Row n=1 of A262586.

BrownE := proc(r,n,m)
    local j,s,p ;
    if r < 1 then
        return 0 ;
    elif r = 1 then
        return A146305(n,m) ;
    elif r = 2 then
        j := n mod 2 ; s := floor(n/2) ;
        if type(m,'even') then
            return 0 ;
        end if;
        p := (m+1)/2 ;
        if p > 0 and s >= 0 then
            return 2*(2*p)!*(4*s+2*p+2*j-1)!/p!/(p-1)!/s!/(3*s+2*p+2*j)! ;
        else
            return 0 ;
        end if;
    elif r =3 and (n mod 3) =0 and (m mod 3) = 0 then
        s := n/3 ; p := m/3 ;
        if p >= 0 and s >= 0 then
            return (2*p+1)!*(4*s+2*p)!/p!/p!/s!/(3*s+2*p+1)! ;
        else
            return 0 ;
        end if;
    elif r >= 3 then
        if ((n-1) mod r) =0 and ((m+3) mod r) =0 then
            s := (n-1)/r ; p := (m+3)/r-1 ;
            if p>=0 and s>=0 then
            return (2*p+2)!*(4*s+2*p+1)!/p!/(p+1)!/s!/(3*s+2*p+2)! ;
            else
                return 0 ;
            end if;
        else
            return 0 ;
        end if;
    else
        return 0 ;
    end if;
end proc:
BrownG := proc(n,m)
    add( numtheory[phi](s)* BrownE(s,n,m), s = numtheory[divisors](m+3) ) ;
    %/(m+3) ;
end proc:
A210696 := proc(n)
    BrownG(1,n) ;
end proc:
seq(A210696(n),n=0..25) ;

Mathematica
```
(* See the link section. *)
```

a(26) onwards from Andrew Howroyd, Nov 23 2024

A378335 Number of n-node connected sensed planar maps with an external face in which all internal faces are triangles.

Original entry on oeis.org

1, 2, 5, 23, 114, 675, 4357, 29692, 209933, 1531349, 11439545, 87202034, 676101375, 5319069978, 42378741675, 341403125204, 2777329727768, 22790623236250, 188475690185517, 1569588986566488, 13153980294670851, 110870960421707325, 939397206702263021

Offset: 3

Andrew Howroyd, Nov 23 2024

nonn

See A377785 for illustration of initial terms. This sequence does not consider a planar map to be equivalent to its mirror image.

Andrew Howroyd, Table of n, a(n) for n = 3..200

Row sums of A378336.

Cf. A262586, A377785, A378334, A378337, A378338, A378339, A378340.

A378335seq(25) \\ See PARI link in A378340 for program code.

A378336 Triangle read by rows: T(n,k) is the number of n node connected sensed planar maps with an external face and k triangular internal faces, n >= 3, 1 <= k <= 2*n - 5.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 2, 1, 0, 0, 2, 5, 5, 6, 5, 0, 0, 2, 8, 13, 20, 21, 26, 24, 0, 0, 0, 10, 28, 55, 79, 104, 119, 147, 133, 0, 0, 0, 7, 45, 126, 230, 360, 491, 625, 735, 892, 846, 0, 0, 0, 0, 44, 227, 561, 1066, 1682, 2430, 3241, 4074, 4830, 5876, 5661

Offset: 3

Andrew Howroyd, Nov 23 2024

See A378103 for illustration of initial terms. This sequence does not consider planar maps to be equivalent to their reflections.
The planar maps considered are without loops or isthmuses.
In other words, a(n) is the number of embeddings in the plane of connected bridgeless planar simple graphs with n vertices and k triangular internal faces up to orientation preserving isomorphisms.
The number of edges is n + k - 1.

			Triangle begins:
n\k | 1  2  3   4   5    6    7    8    9   10   11   12   13
----+----------------------------------------------------------
  3 | 1;
  4 | 0, 1, 1;
  5 | 0, 1, 1,  2,  1;
  6 | 0, 0, 2,  5,  5,   6,   5;
  7 | 0, 0, 2,  8, 13,  20,  21,  26,  24;
  8 | 0, 0, 0, 10, 28,  55,  79, 104, 119, 147, 133;
  9 | 0, 0, 0,  7, 45, 126, 230, 360, 491, 625, 735, 892, 846;
  ...

Andrew Howroyd, Table of n, a(n) for n = 3..2306 (rows 3..50)

Row sums are A378335.
Column sums are A378337.
Antidiagonal sums are A378338.
The final 3 terms of each row are in A002709, A002710, A002711.
Cf. A262586 (2-connected), A341923 (3-connected), A378103, (unsensed), A378340 (achiral).

my(A=A378336rows(10)); for(i=1, #A, print(A[i])) \\ See PARI link in A378340 for program code.

T(n,k) = 0 for n > 2*k + 1.
T(n,2*n-5) = A002709(n-3).
T(n,2*n-6) = A002710(n-4) for n >= 4.
T(n,2*n-7) = A002711(n-5) for n >= 5.

A002710 Triangulations of the disk G_{n,1}.

Original entry on oeis.org

1, 2, 6, 26, 147, 892, 5876, 40490, 290020, 2136488, 16113254, 123878966, 968017330, 7670113856, 61510346760, 498496979754, 4077605379276, 33629943832280, 279413323740280, 2336935584712872, 19663001667901339, 166348460274745684, 1414318445894183076, 12079654921382780966

Offset: 0

N. J. A. Sloane

nonn

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).

Andrew Howroyd, Table of n, a(n) for n = 0..500
William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768.
William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768. [Annotated scanned copy]

Column k=1 of A262586.

Extended by Max Alekseyev, Mar 30 2009

a(21) onwards from Andrew Howroyd, Nov 23 2024

cp's OEIS Frontend

A001683 Number of one-sided triangulations of the disk; or flexagons of order n; or unlabeled plane trivalent trees (n-2 internal vertices, all of degree 3 and hence n leaves).

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A002709 Triangulations of the disk G_{n,0}.

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A169808 Array T(n,k) read by antidiagonals: T(n,k) is the number of [n,k]-triangulations in the plane, n >= 0, k >= 0.

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A169809 Array T(n,k) read by antidiagonals: T(n,k) is the number of [n,k]-triangulations in the plane that have reflection symmetry, n >= 0, k >= 0.

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A146305 Array T(n,m) = 2*(2m+3)!*(4n+2m+1)!/(m!*(m+2)!*n!*(3n+2m+3)!) read by antidiagonals.

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A341923 Array read by antidiagonals: T(n,k) is the number of 3-connected triangulations of a disk up to orientation-preserving isomorphisms with n interior nodes and k nodes on the boundary, n >= 1, k >= 3.

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A210696 Triangulations of the disk, G_{1,n}.

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A146305 Array T(n,m) = 2(2m+3)!(4n+2m+1)!/(m!(m+2)!n!*(3n+2m+3)!) read by antidiagonals.