A295260 -id:A295260 - cp's OEIS frontend

A000207 Number of inequivalent ways of dissecting a regular (n+2)-gon into n triangles by n-1 non-intersecting diagonals under rotations and reflections; also the number of (unlabeled) maximal outerplanar graphs on n+2 vertices.

1, 1, 1, 3, 4, 12, 27, 82, 228, 733, 2282, 7528, 24834, 83898, 285357, 983244, 3412420, 11944614, 42080170, 149197152, 531883768, 1905930975, 6861221666, 24806004996, 90036148954, 327989004892, 1198854697588, 4395801203290, 16165198379984, 59609171366326, 220373278174641

Offset: 1

N. J. A. Sloane

Also a(n) is the number of hexaflexagons of order n+2. - Mike Godfrey (m.godfrey(AT)umist.ac.uk), Feb 25 2002 (see the Kosters paper).
Number of normally non-isomorphic realizations of the associahedron of type II with dimension n in Ceballos et al. - Tom Copeland, Oct 19 2011
Number of polyforms with n cells in the hyperbolic tiling with Schläfli symbol {3,oo}, not distinguishing enantiomorphs. - Thomas Anton, Jan 16 2019
A stereographic projection of the {3,oo} tiling on the Poincaré disk can be obtained via the Christensson link. - Robert A. Russell, Jan 20 2024
A maximal outerplanar graph (MOP) has a plane embedding with all vertices on the exterior region and interior regions triangles. - Allan Bickle, Feb 25 2024

			E.g., a square (4-gon, n=2) could have either diagonal drawn, C(3)=2, but with essentially only one result. A pentagon (5-gon, n=3) gives C(4)=5, but they each have 2 diags emanating from 1 of the 5 vertices and are essentially the same. A hexagon can have a nuclear disarmament sign (6 ways), an N (3 ways and 3 reflections) or a triangle (2 ways) of diagonals, 6 + 6 + 2 = 14 = C(5), but only 3 essentially different. - _R. K. Guy_, Mar 06 2004
G.f. = x + x^2 + x^3 + 3*x^4 + 4*x^5 + 12*x^6 + 27*x^7 + 82*x^8 + ...

L. W. Beineke and R. E. Pippert, Enumerating labeled k-dimensional trees and ball dissections, pp. 12-26 of Proceedings of Second Chapel Hill Conference on Combinatorial Mathematics and Its Applications, University of North Carolina, Chapel Hill, 1970. Reprinted in Math. Annalen, 191 (1971), 87-98.
Cameron, Peter J. Some treelike objects. Quart. J. Math. Oxford Ser. (2) 38 (1987), no. 150, 155--183. MR0891613 (89a:05009). See pp. 155, 163, but note that the formulas on p. 163, lines 5 and 6, contain typos. See the correct formulas given here. - N. J. A. Sloane, Apr 18 2014
B. N. Cyvin, E. Brendsdal, J. Brunvoll and S. J. Cyvin, Isomers of polyenes attached to benzene, Croatica Chemica Acta, 68 (1995), 63-73.
S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751.
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.
R. K. Guy, "Dissecting a polygon into triangles," Bull. Malayan Math. Soc., Vol. 5, pp. 57-60, 1958.
R. K. Guy, Dissecting a polygon into triangles, Research Paper #9, Math. Dept., Univ. Calgary, 1967.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 79, Table 3.5.1 (the entries for n=16 and n=21 appear to be incorrect).
M. Kosters, A theory of hexaflexagons, Nieuw Archief Wisk., 17 (1999), 349-362.
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).
P. K. Stockmeyer, The charm bracelet problem and its applications, pp. 339-349 of Graphs and Combinatorics (Washington, Jun 1973), Ed. by R. A. Bari and F. Harary. Lect. Notes Math., Vol. 406. Springer-Verlag, 1974.

T. D. Noe, Table of n, a(n) for n = 1..200
F. R. Bernhart & N. J. A. Sloane, Correspondence, 1977
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
Douglas Bowman and Alon Regev, Counting symmetry classes of dissections of a convex regular polygon, arXiv preprint arXiv:1209.6270 [math.CO], 2012.
William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
P. J. Cameron, Some treelike objects, Quart. J. Math. Oxford, 38 (1987), 155-183. See p. 160.
C. Ceballos, F. Santos, and G. Ziegler, Many Non-equivalent Realizations of the Associahedron, arXiv:1109.5544 [math.MG], 2011-2013, p. 19 and 26.
Malin Christensson, Make hyperbolic tilings of images, web page, 2019.
Sean Cleary, Roland Maio, Counting difficult tree pairs with respect to the rotation distance problem, arXiv:2001.06407 [cs.DS], 2020.
A. S. Conrad and D. K. Hartline, Flexagons
S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743-751. [Annotated scanned copy]
R. K. Guy, Dissecting a polygon into triangles, Research Paper #9, Math. Dept., Univ. Calgary, 1967. [Annotated scanned copy]
F. Harary and E. M. Palmer, On acyclic simplicial complexes, Mathematika 15 1968 115-122.
F. Harary, E. M. Palmer, and 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 (the entries for n=4 and n=30 appear to be incorrect).
J. W. Moon and L. Moser, Triangular dissections of n-gons, Canad. Math. Bull., 6 (1963), 175-178.
T. Motzkin, The hypersurface cross ratio, Bull. Amer. Math. Soc., 51 (1945), 976-984.
T. S. Motzkin, Relations between hypersurface cross ratios and a combinatorial formula for partitions of a polygon, for permanent preponderance and for non-associative products, Bull. Amer. Math. Soc., 54 (1948), 352-360 (the entry for n=10 appears to be incorrect).
C. O. Oakley and R. J. Wisner, Flexagons, Amer. Math. Monthly 64 (1957), 143-154.
Hans Rademacher, On the number of certain types of polyhedra, Illinois Journal of Mathematics 9.3 (1965): 361-380. Reprinted in Coll. Papers, Vol II, MIT Press, 1974, pp. 544-564.
Manfred Scheucher, Hendrik Schrezenmaier, Raphael Steiner, A Note On Universal Point Sets for Planar Graphs, arXiv:1811.06482 [math.CO], 2018.
Len Smiley, Illustration of initial terms
Tiberiu Spircu and Stefan V. Pantazi, Again around frieze patterns, arXiv:2002.08211 [math.CO], 2020. See Kn p. 13.
P. J. Stockmeyer, The charm bracelet problem and its applications, pp. 339-349 of Graphs and Combinatorics (Washington, Jun 1973), Ed. by R. A. Bari and F. Harary. Lect. Notes Math., Vol. 406. Springer-Verlag, 1974. [Scanned annotated and corrected copy]

Column k=3 of A295260.
Cf. A000577, A070765.
A row or column of the array in A169808.
Polyominoes: A001683(n+2) (oriented), A369314 (chiral), A208355(n-1) (achiral), A005036 {4,oo}, A007173 {3,3,oo}.
Cf. A097998, A097999, A098000 (labeled outerplanar graphs).
Cf. A111563, A111564, A111758, A111759, A111757 (unlabeled outerplanar graphs).

A000108 := proc(n) if n >= 0 then binomial(2*n,n)/(n+1) ; else 0; fi; end:
A000207 := proc(n) option remember: local k, it1, it2;
if n mod 2 = 0 then k := n/2+2 else k := (n+3)/2 fi:
if n mod 2 <> 0 then it1 := 0 else it1 := 1 fi:
if (n+2) mod 3 <> 0 then it2 := 0 else it2 := 1 fi:
RETURN(A000108(n)/(2*n+4) + it1*A000108(n/2)/4 + A000108(k-2)/2 + it2*A000108((n-1)/3)/3)
end:
seq(A000207(n),n=1..30) ; # (Revised Maple program from R. J. Mathar, Apr 19 2009)
A000207 := proc(n) option remember: local k,it1,it2; if n mod 2 = 0 then k := n/2+1 else k := (n+1)/2 fi: if n mod 2 <> 0 then it1 := 0 else it1 := 1 fi: if n mod 3 <> 0 then it2 := 0 else it2 := 1 fi: RETURN(A000108(n-2)/(2*n) + it1*A000108(n/2+1-2)/4 + A000108(k-2)/2 + it2*A000108(n/3+1-2)/3) end:
A000207 := n->(A000108(n)/(n+2)+A000108(floor(n/2))*((1+(n+1 mod 2) /2)))/2+`if`(n mod 3=1,A000108(floor((n-1)/3))/3,0); # Peter Luschny, Apr 19 2009 and M. F. Hasler, Apr 19 2009
G:=(12*(1+x-2*x^2)+(1-4*x)^(3/2)-3*(3+2*x)*(1-4*x^2)^(1/2)-4*(1-4*x^3)^(1/2))/24/x^2: Gser:=series(G,x=0,35): seq(coeff(Gser,x^n),n=1..31); # Emeric Deutsch, Dec 19 2004

p=3; Table[(Binomial[(p-1)n, n]/(((p-2)n+1)((p-2)n+2)) + If[OddQ[n], If[OddQ[p], Binomial[(p-1)n/2, (n-1)/2]/n, (p+1)Binomial[((p-1)n-1)/2, (n-1)/2]/((p-2)n+2)], 3Binomial[(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, 2}]])/2, {n, 1, 20}] (* Robert A. Russell, Dec 11 2004 *)
a[n_] := (CatalanNumber[n]/(n+2) + CatalanNumber[ Quotient[n, 2]] *((1 + Mod[n-1, 2]/2)))/2 + If[Mod[n, 3] == 1, CatalanNumber[ Quotient[n-1, 3]]/3, 0] ; Table[a[n], {n, 1, 28}] (* Jean-François Alcover, Sep 08 2011, after PARI *)

A000207(n)=(A000108(n)/(n+2)+A000108(n\2)*if(n%2,1,3/2))/2+if(n%3==1,A000108(n\3)/3) \\ M. F. Hasler, Apr 19 2009

a(n) = C(n)/(2*n) + C(n/2+1)/4 + C(k)/2 + C(n/3+1)/3 where C(n) = A000108(n-2) if n is an integer, 0 otherwise and k = (n+1)/2 if n is odd, k = n/2+1 if n is even. Thus C(2), C(3), C(4), C(5), ... are 1, 1, 2, 5, ...
G.f.: (12*(1+x-2*x^2) + (1-4*x)^(3/2) - 3*(3+2*x)*(1-4*x^2)^(1/2) - 4*(1-4*x^3)^(1/2))/(24*x^2). - Emeric Deutsch, Dec 19 2004, from the S. J. Cyvin et al. reference.
a(n) ~ A000108(n)/(2*n+4) ~ 4^n / (2 sqrt(n Pi)*(n + 1)*(n + 2)). - M. F. Hasler, Apr 19 2009
a(n) = A001683(n+2) - A369314(n) = (A001683(n+2) + A208355(n-1)) / 2 = A369314(n) + A208355(n-1). - Robert A. Russell, Jan 19 2024
Beineke and Pippert have an explicit formula with six cases (based on the value of n mod 6). - Allan Bickle, Feb 25 2024

More terms from James Sellers, Jul 10 2000

A001004 Number of nonequivalent dissections of an (n+2)-gon by nonintersecting diagonals up to rotation and reflection.

Original entry on oeis.org

1, 1, 2, 3, 9, 20, 75, 262, 1117, 4783, 21971, 102249, 489077, 2370142, 11654465, 57916324, 290693391, 1471341341, 7504177738, 38532692207, 199076194985, 1034236705992, 5400337050086, 28329240333758, 149244907249629

Offset: 0

N. J. A. Sloane

Original name: number of symmetric dissections of a polygon.

Also number of 2-connected outerplanar graphs on n unlabeled nodes. - Steven Finch, Dec 09 2004

Cameron, Peter J. Some treelike objects. Quart. J. Math. Oxford Ser. (2) 38 (1987), no. 150, 155--183. MR0891613 (89a:05009). See p. 155. - N. J. A. Sloane, Apr 18 2014
Guanzhang Hu, Group theory method for enumeration of outerplanar graphs, Acta Math. Appl. Sinica 14 (1998) 381-387.
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..200
S. R. Finch, Planar graph growth constants.
Steven R. Finch, Planar graph growth constants [Cached copy, with permission of the author]
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.
P. Lisonek, Closed forms for the number of polygon dissections, Journal of Symbolic Computation 20 (1995), 595-601.
T. Motzkin, The hypersurface cross ratio, Bull. Amer. Math. Soc., 51 (1945), 976-984.
T. S. Motzkin, Relations between hypersurface cross ratios and a combinatorial formula for partitions of a polygon, for permanent preponderance and for non-associative products, Bull. Amer. Math. Soc., 54 (1948), 352-360.
R. C. Read, On general dissections of a polygon, Preprint (1974)
Ronald C. Read, On general dissections of a polygon, Aequat. math. 18 (1978) 370-388.
James Tilley, Stan Wagon, and Eric Weisstein, A Catalog of Facially Complete Graphs, arXiv:2409.11249 [math.CO], 2024. See pp. 7,11.

Cf. A003454, A003455, A003456, A005036, A295260.

f[x_, n_]:=x+Sum[(1/r)*Binomial[s-2, r-1]*Binomial[r+s-1, s]*x^s, {r, 1, n}, {s, 2, n}]; F[x_, n_]:=Series[((3x^2-2*x*f[x, n]+f[x, n]^2)- (2+2*x+7*x^2-4*x*f[x, n]+2*f[x, n]^2)*f[x^2, n]+ 2*f[x^2, n]^2)/(4*(2*f[x^2, n]-1))+Sum[If[Mod[k, d]==0, EulerPhi[d]*f[x^d, n]^(k/d)/k, 0], {k, 3, n}, {d, 1, k}]/2, {x, 0, n}]; F[x, 22] (Finch)

\\ See A295419 for DissectionsModDihedral().
my(v=DissectionsModDihedral(apply(i->1, [1..30])));v[3..#v] \\ Andrew Howroyd, Nov 22 2017

More terms from Esa Peuha (esa.peuha(AT)helsinki.fi), Oct 21 2005

Name clarified by Andrew Howroyd, Nov 22 2017

A070914 Array read by antidiagonals giving number of paths up and left from (0,0) to (n,kn) where x/y <= k for all intermediate points.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 5, 1, 1, 1, 4, 12, 14, 1, 1, 1, 5, 22, 55, 42, 1, 1, 1, 6, 35, 140, 273, 132, 1, 1, 1, 7, 51, 285, 969, 1428, 429, 1, 1, 1, 8, 70, 506, 2530, 7084, 7752, 1430, 1, 1, 1, 9, 92, 819, 5481, 23751, 53820, 43263, 4862, 1, 1, 1, 10, 117, 1240

Offset: 0

Henry Bottomley, May 20 2002

Also related to dissections of polygons and enumeration of trees.
Number of dissections of a polygon into n (k+2)-gons by nonintersecting diagonals. All dissections are counted separately. See A295260 for nonequivalent solutions up to rotation and reflection. - Andrew Howroyd, Nov 20 2017
Number of rooted polyominoes composed of n (k+2)-gonal cells of the hyperbolic (Euclidean for k=0) regular tiling with Schläfli symbol {k+2,oo}. A rooted polyomino has one external edge identified, and chiral pairs are counted as two. For k>0, a stereographic projection of the {k+2,oo} tiling on the Poincaré disk can be obtained via the Christensson link. - Robert A. Russell, Jan 27 2024

			Rows start:
===========================================================
n\k| 0     1      2       3        4        5         6
---|-------------------------------------------------------
0  | 1,    1,     1,      1,       1,       1,        1 ...
1  | 1,    1,     1,      1,       1,       1,        1 ...
2  | 1,    2,     3,      4,       5,       6,        7 ...
3  | 1,    5,    12,     22,      35,      51,       70 ...
4  | 1,   14,    55,    140,     285,     506,      819 ...
5  | 1,   42,   273,    969,    2530,    5481,    10472 ...
6  | 1,  132,  1428,   7084,   23751,   62832,   141778 ...
7  | 1,  429,  7752,  53820,  231880,  749398,  1997688 ...
8  | 1, 1430, 43263, 420732, 2330445, 9203634, 28989675 ...
...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Malin Christensson, Make hyperbolic tilings of images, web page, 2019.
Peter Hilton and Jean Pedersen, Catalan Numbers, Their Generalization, and Their Uses, The Mathematical Intelligencer, March 1991, Volume 13, Issue 2, pp. 64-75.
V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., 14 (1976), 395-405.

Rows include A000012 (twice), A000027, A000326.
Columns include A000012, A000108 (Catalan), A001764, A002293, A002294, A002295, A002296, A007556, A062994, A059968.
Reflected version of A062993 (which is the main entry).
Cf. A295260.
Polyominoes: A295224 (oriented), A295260 (unoriented).

A:= (n, k)-> binomial((k+1)*n, n)/(k*n+1):
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Mar 25 2015

T[n_, k_] = Binomial[n(k+1), n]/(k*n+1); Flatten[Table[T[n-k, k], {n, 0, 9}, {k, n, 0, -1}]] (* Jean-François Alcover, Apr 08 2016 *)

T(n, k) = binomial(n*(k+1), n)/(n*k+1); \\ Andrew Howroyd, Nov 20 2017

T(n, k) = binomial(n*(k+1), n)/(n*k+1) = A071201(n, k*n) = A071201(n, k*n+1) = A071202(n, k*n+1) = A062993(n+k-1, k-1).

If P(k,x) = Sum_{n>=0} T(n,k)*x^n is the g.f. of column k (k>=0), then P(k,x) = exp(1/(k+1)*(Sum_{j>0} (1/j)*binomial((k+1)*j,j)*x^j)). - Werner Schulte, Oct 13 2015

A005036 Number of nonequivalent dissections of a polygon into n quadrilaterals by nonintersecting diagonals up to rotation and reflection.

Original entry on oeis.org

1, 1, 2, 5, 16, 60, 261, 1243, 6257, 32721, 175760, 963900, 5374400, 30385256, 173837631, 1004867079, 5861610475, 34469014515, 204161960310, 1217145238485, 7299007647552, 44005602441840

Offset: 1

N. J. A. Sloane

Closed formula is given in my paper linked below. - Nikos Apostolakis, Aug 01 2018

Number of unoriented polyominoes composed of n square cells of the hyperbolic regular tiling with Schläfli symbol {4,oo}. A stereographic projection of this tiling on the Poincaré disk can be obtained via the Christensson link. For unoriented polyominoes, chiral pairs are counted as one. - Robert A. Russell, Jan 20 2024

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 = 1..100
Nikos Apostolakis, Non-crossing trees, quadrangular dissections, ternary trees, and duality preserving bijections, arXiv:1807.11602 [math.CO], July 2018.
Malin Christensson, Make hyperbolic tilings of images, web page, 2019.
F. Harary, E. M. Palmer, and 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. V. Konstantinova, A survey of the cell-growth problem and some its variations, Com 2 MaC-KOSEF, 2001. (Archived link.)
Index entries for "core" sequences

Column k=4 of A295260.
Cf. A005419, A004127.
Polyominoes: A005034 (oriented), A369315 (chiral), A047749 (achiral), A385149 (asymmetric), A001764 (rooted), A000207 {3,oo}, A005040 {5,oo}, A005038 {5,oo} (oriented).

p=4; Table[(Binomial[(p-1)n, n]/(((p-2)n+1)((p-2)n+2)) + If[OddQ[n], If[OddQ[p], Binomial[(p-1)n/2, (n-1)/2]/n, (p+1)Binomial[((p-1)n-1)/2, (n-1)/2]/((p-2)n+2)], 3Binomial[(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, 2}]])/2, {n, 1, 20}] (* Robert A. Russell, Dec 11 2004 *)
Table[(3Binomial[3n,n]/(2n+1)-Binomial[3n+1,n]/(n+1)-If[OddQ[n],-10Binomial[(3n-1)/2,(n-1)/2]-If[1==Mod[n,4],4Binomial[(3n-3)/4,(n-1)/4],0],-6Binomial[3n/2,n/2]]/(n+1))/8,{n,0,30}] (* Robert A. Russell, Jun 19 2025 *)

a(n) ~ 3^(3*n + 1/2) / (sqrt(Pi) * n^(5/2) * 2^(2*n + 4)). - Vaclav Kotesovec, Mar 13 2016
a(n) = A005034(n) - A369315(n) = (A005034(n) + A047749(n)) / 2 = A369315(n) + A047749(n). - Robert A. Russell, Jan 19 2024
G.f.: (3*G(z) - G(z)^2 + 6*G(z^2) + 5z*G(z^2)^2 + 2z*G(z^4)) / 8, where G(z)=1+z*G(z)^3 is the g.f. for A001764. - Robert A. Russell, Jun 19 2025

More terms from Sascha Kurz, Oct 13 2001

Name edited by Andrew Howroyd, Nov 20 2017

A295224 Array read by antidiagonals: T(n,k) = number of nonequivalent dissections of a polygon into n k-gons by nonintersecting diagonals up to rotation (k >= 3).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 1, 1, 2, 7, 6, 1, 1, 3, 12, 25, 19, 1, 1, 3, 19, 57, 108, 49, 1, 1, 4, 26, 118, 366, 492, 150, 1, 1, 4, 35, 203, 931, 2340, 2431, 442, 1, 1, 5, 46, 332, 1989, 7756, 16252, 12371, 1424, 1, 1, 5, 57, 494, 3766, 20254, 68685, 115940, 65169, 4522

Offset: 1

Andrew Howroyd, Nov 17 2017

The polygon prior to dissection will have n*(k-2)+2 sides.
In the Harary, Palmer and Read reference these are the sequences called H.
T(n,k) is the number of oriented polyominoes containing n k-gonal tiles of the hyperbolic regular tiling with Schläfli symbol {k,oo}. Stereographic projections of several of these tilings on the Poincaré disk can be obtained via the Christensson link. For oriented polyominoes, chiral pairs are counted as two. T(n,2) could represent polyominoes of the Euclidean regular tiling with Schläfli symbol {2,oo}; T(n,2) = 1. - Robert A. Russell, Jan 21 2024

			Array begins:
  =====================================================
  n\k|    3     4      5       6        7         8
  ---|-------------------------------------------------
   1 |    1     1      1       1        1         1 ...
   2 |    1     1      1       1        1         1 ...
   3 |    1     2      2       3        3         4 ...
   4 |    4     7     12      19       26        35 ...
   5 |    6    25     57     118      203       332 ...
   6 |   19   108    366     931     1989      3766 ...
   7 |   49   492   2340    7756    20254     45448 ...
   8 |  150  2431  16252   68685   219388    580203 ...
   9 |  442 12371 115940  630465  2459730   7684881 ...
  10 | 1424 65169 854981 5966610 28431861 104898024 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275
Malin Christensson, Make hyperbolic tilings of images, web page, 2019.
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.
Wikipedia, Fuss-Catalan number

Columns k=3..6 are A001683(n+2), A005034, A005038, A221184(n-1).
Cf. A033282, A070914, A295222, A295259, A295260.
Polyominoes: A295260 (unoriented), A070914 (rooted).

u[n_, k_, r_] := r*Binomial[(k - 1)*n + r, n]/((k - 1)*n + r);
T[n_, k_] := u[n, k, 1] + (If[EvenQ[n], u[n/2, k, 1], 0] - u[n, k, 2])/2 + DivisorSum[GCD[n - 1, k], EulerPhi[#]*u[(n - 1)/#, k, k/#]&]/k;
Table[T[n - k + 1, k], {n, 1, 13}, {k, n, 3, -1}] // Flatten (* Jean-François Alcover, Nov 21 2017, after Andrew Howroyd *)

\\ here u is Fuss-Catalan sequence with p = k+1.
u(n, k, r)={r*binomial((k - 1)*n + r, n)/((k - 1)*n + r)}
T(n,k) = u(n,k,1) + (if(n%2==0, u(n/2,k,1))-u(n,k,2))/2 + sumdiv(gcd(n-1,k), d, eulerphi(d)*u((n-1)/d,k,k/d))/k;
for(n=1, 10, for(k=3, 8, print1(T(n, k), ", ")); print);

from sympy import binomial, gcd, totient, divisors
def u(n, k, r): return r*binomial((k - 1)*n + r, n)//((k - 1)*n + r)
def T(n, k): return u(n, k, 1) + ((u(n//2, k, 1) if n%2==0 else 0) - u(n, k, 2))//2 + sum([totient(d)*u((n - 1)//d, k, k//d) for d in divisors(gcd(n - 1, k))])//k
for n in range(1, 11): print([T(n, k) for k in range(3, 9)]) # Indranil Ghosh, Dec 13 2017, after PARI code

T(n,k) ~ A295222(n,k)/n for fixed k.

A004127 Number of planar hexagon trees with n hexagons.

Original entry on oeis.org

1, 1, 3, 12, 68, 483, 3946, 34485, 315810, 2984570, 28907970, 285601251, 2868869733, 29227904840, 301430074416, 3141985563575, 33059739636198, 350763452126835, 3749420616902637, 40348040718155170, 436827335493148600

Offset: 1

N. J. A. Sloane

nonn

Number of nonequivalent dissections of a polygon into n hexagons by nonintersecting diagonals up to rotation and reflection. - Andrew Howroyd, Nov 20 2017

Number of unoriented polyominoes composed of n hexagonal cells of the hyperbolic regular tiling with Schläfli symbol {6,oo}. A stereographic projection of this tiling on the Poincaré disk can be obtained via the Christensson link. For unoriented polyominoes, chiral pairs are counted as one. - Robert A. Russell, Jan 23 2024

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

G. C. Greubel, Table of n, a(n) for n = 1..925
Malin Christensson, Make hyperbolic tilings of images, web page, 2019.
L. W. Beineke and R. E. Pippert, On the enumeration of planar trees of hexagons, Glasgow Math. J., 15 (1974), 131-147.
L. W. Beineke and R. E. Pippert, On the enumeration of planar trees of hexagons, Glasgow Math. J., 15 (1974), 131-147. [Annotated scanned copy]
Index entries for sequences related to trees

Column k=6 of A295260.
Cf. A002294.
Polyominoes: A221184{n-1} (oriented), A369473 (chiral), A143546 (achiral), A005040 {5,oo}, A005419 {7,oo}.

T := proc(n) if floor(n)=n then binomial(5*n+1,n)/(5*n+1) else 0 fi end: U := proc(n) if n mod 2 = 0 then binomial(5*n/2+1, n/2)/(5*n/2+1) else 6*binomial((5*n+1)/2,(n-1)/2)/(5*n+1) fi end: S := n->T(n)/4/(2*n+1)+T(n/2)/6+(5*n-2)*T((n-1)/3)/6/(2*n+1)+T((n-1)/6)/6+7*U(n)/12: seq(S(n),n=1..25); (Emeric Deutsch)

p=6; Table[(Binomial[(p-1)n, n]/(((p-2)n+1)((p-2)n+2)) + If[OddQ[n], If[OddQ[p], Binomial[(p-1)n/2, (n-1)/2]/n, (p+1)Binomial[((p-1)n-1)/2, (n-1)/2]/((p-2)n+2)], 3Binomial[(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, 2}]])/2, {n, 1, 20}] (* Robert A. Russell, Dec 11 2004 *)

See Theorem 3 on p. 142 in the Beineke-Pippert paper; also the Maple and Mathematica codes here.
a(n) ~ 5^(5*n + 1/2) / (sqrt(Pi) * n^(5/2) * 2^(8*n + 13/2)). - Vaclav Kotesovec, Mar 13 2016
a(n) = A221184(n-1) - A369473(n) = (A221184(n-1) + A143546(n)) / 2 = A369473(n) + A143546(n). - Robert A. Russell, Jan 23 2024

More terms from Emeric Deutsch, Jan 22 2004

A295419 Number of dissections of an n-gon by nonintersecting diagonals into polygons with a prime number of sides counted up to rotations and reflections.

Original entry on oeis.org

1, 1, 2, 4, 8, 23, 64, 222, 752, 2805, 10475, 40614, 158994, 633456, 2548241, 10362685, 42485242, 175557329, 730314350, 3056971164, 12867007761, 54434131848, 231354091945, 987496927875, 4231561861914, 18198894300129, 78533356685275, 339958801585826

Offset: 3

Andrew Howroyd, Nov 22 2017

nonn

a(n) first differs from A290816(n) at n=9 since this sequence does not allow the trivial dissection of a nonagon into a single nonagon.

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

Cf. A001004, A290571, A290646, A290722, A290816, A295260, A295634.

DissectionsModDihedral[v_] := Module[{n = Length[v], q, vars, u, R, Q, T, p}, q = Table[0, {n}]; q[[1]] = InverseSeries[x - Sum[x^i v[[i]], {i, 3, Length[v]}]/x + O[x]^(n+1)]; For[i = 2, i <= n, i++, q[[i]] = q[[i-1]] q[[1]]]; vars = Variables[q[[1]]]; u[m_, r_] := Normal[(q[[r]] + O[x]^(Quotient[n, m]+1))] /. Thread[vars -> vars^m]; R = Sum[v[[2i+1]] u[2, i], {i, 1, (Length[v]-1)/2 // Floor}]; Q = Sum[v[[2i]] u[2, i-1], {i, 2, Length[v]/2 // Floor}]; T = Sum[v[[i]] Sum[EulerPhi[d] u[d, i/d], {d, Divisors[i]}]/i, {i, 3, Length[v]}]; p = O[x]^n - x^2 + (x u[1, 1] + u[2, 1] + (Q u[2, 1] - u[1, 2] + (x+R)^2/(1-Q))/2 + T)/2; Drop[ CoefficientList[p, x], 3]];
DissectionsModDihedral[Boole[PrimeQ[#]]& /@ Range[1, 31]] (* Jean-François Alcover, Sep 25 2019, after Andrew Howroyd *)

\\ number of dissections into parts defined by set.
DissectionsModDihedral(v)={my(n=#v);
my(q=vector(n)); q[1]=serreverse(x-sum(i=3,#v,x^i*v[i])/x + O(x*x^n));
for(i=2, n, q[i]=q[i-1]*q[1]);
my(vars=variables(q[1]));
my(u(m, r)=substvec(q[r]+O(x^(n\m+1)), vars, apply(t->t^m, vars)));
my(R=sum(i=1, (#v-1)\2, v[2*i+1]*u(2,i)), Q=sum(i=2, #v\2, v[2*i]*u(2,i-1)), T=sum(i=3, #v, my(c=v[i]); if(c, c*sumdiv(i, d, eulerphi(d)*u(d,i/d))/i)));
my(p=O(x*x^n) - x^2 + (x*u(1,1) + u(2,1) + (Q*u(2,1) - u(1,2) + (x+R)^2/(1-Q))/2 + T)/2);
vector(n,i,polcoeff(p,i))}
DissectionsModDihedral(apply(v->isprime(v), [1..25]))

A005040 Number of nonequivalent dissections of a polygon into n pentagons by nonintersecting diagonals up to rotation and reflection.

Original entry on oeis.org

1, 1, 2, 8, 33, 194, 1196, 8196, 58140, 427975, 3223610, 24780752, 193610550, 1534060440, 12302123640, 99699690472, 815521503060, 6725991120004, 55882668179880, 467387136083296, 3932600361607809, 33269692212847056, 282863689410850236, 2415930985594609548

Offset: 1

N. J. A. Sloane

nonn

Number of unoriented polyominoes composed of n pentagonal cells of the hyperbolic regular tiling with Schläfli symbol {5,oo}. A stereographic projection of this tiling on the Poincaré disk can be obtained via the Christensson link. For unoriented polyominoes, chiral pairs are counted as one. - Robert A. Russell, Jan 23 2024

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

G. C. Greubel, Table of n, a(n) for n = 1..1000
Malin Christensson, Make hyperbolic tilings of images, web page, 2019.
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. V. Konstantinova, A survey of the cell-growth problem and some its variations, preprint, 2001.
E. V. Konstantinova, Com2Mac - Preprints [Dead link?]

Column k=5 of A295260.

Polyominoes: A005038 (oriented), A369471 (chiral), A369472 (achiral), A000207 {3,oo}, A005036 {4,oo}, A004127 {6,oo}, A005419 {7,oo}.

p=5; Table[(Binomial[(p-1)n, n]/(((p-2)n+1)((p-2)n+2)) + If[OddQ[n], If[OddQ[p], Binomial[(p-1)n/2, (n-1)/2]/n, (p+1)Binomial[((p-1)n-1)/2, (n-1)/2]/((p-2)n+2)], 3Binomial[(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, 2}]])/2, {n, 1, 20}] (* Robert A. Russell, Dec 11 2004 *)

See Mathematica code.
a(n) ~ 2^(8*n - 1/2) / (sqrt(Pi) * n^(5/2) * 3^(3*n + 5/2)). - Vaclav Kotesovec, Mar 13 2016
a(n) = A005038(n) - A369471(n) = (A005038(n) + A369472(n)) / 2 = A369471(n) + A369472(n). - Robert A. Russell, Jan 23 2024

More terms from Sascha Kurz, Oct 13 2001.

Name edited by Andrew Howroyd, Nov 20 2017.

A295222 Array read by antidiagonals: T(n,k) is the number of nonequivalent dissections of a polygon into n k-gons by nonintersecting diagonals rooted at a cell up to rotation (k >= 3).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 5, 10, 1, 1, 6, 22, 30, 1, 1, 8, 40, 116, 99, 1, 1, 9, 64, 285, 612, 335, 1, 1, 11, 92, 578, 2126, 3399, 1144, 1, 1, 12, 126, 1015, 5481, 16380, 19228, 3978, 1, 1, 14, 166, 1641, 11781, 54132, 129456, 111041, 14000

Offset: 1

Andrew Howroyd, Nov 17 2017

The polygon prior to dissection will have n*(k-2)+2 sides.

In the Harary, Palmer and Read reference these are the sequences called F.

			Array begins:
  ===========================================================
  n\k|     3      4       5        6         7          8
  ---|-------------------------------------------------------
   1 |     1      1       1        1         1          1 ...
   2 |     1      1       1        1         1          1 ...
   3 |     3      5       6        8         9         11 ...
   4 |    10     22      40       64        92        126 ...
   5 |    30    116     285      578      1015       1641 ...
   6 |    99    612    2126     5481     11781      22386 ...
   7 |   335   3399   16380    54132    141778     317860 ...
   8 |  1144  19228  129456   548340   1753074    4638348 ...
   9 |  3978 111041 1043460  5672645  22137570   69159400 ...
  10 | 14000 650325 8544965 59653210 284291205 1048927880 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275
F. Harary, E. M. Palmer and R. C. Read, On the cell-growth problem for arbitrary polygons, Discr. Math. 11 (1975), 371-389.
Wikipedia, Fuss-Catalan number

Columns k=3..5 are A003441, A005033, A005037.

Cf. A033282, A070914, A295224, A295259, A295260.

u[n_, k_, r_] := r*Binomial[(k - 1)*n + r, n]/((k - 1)*n + r);
T[n_, k_] := DivisorSum[GCD[n-1, k], EulerPhi[#]*u[(n-1)/#, k, k/#]&]/k;
Table[T[n - k + 3, k], {n, 1, 10}, {k, n + 2, 3, -1}] // Flatten (* Jean-François Alcover, Nov 21 2017, after Andrew Howroyd *)

\\ here u is Fuss-Catalan sequence with p = k+1.
u(n,k,r)={r*binomial((k - 1)*n + r, n)/((k - 1)*n + r)}
T(n,k)=sumdiv(gcd(n-1,k), d, eulerphi(d)*u((n-1)/d, k, k/d))/k;
for(n=1, 10, for(k=3, 8, print1(T(n, k), ", ")); print);

from sympy import binomial, gcd, totient, divisors
def u(n, k, r): return r*binomial((k - 1)*n + r, n)//((k - 1)*n + r)
def T(n, k): return sum([totient(d)*u((n - 1)//d, k, k//d) for d in divisors(gcd(n - 1, k))])//k
for n in range(1, 11): print([T(n, k) for k in range(3, 9)]) # Indranil Ghosh, Dec 13 2017, after PARI

T(n,k) = Sum_{d|gcd(n-1,k)} phi(d)*u((n-1)/d, k, k/d)/k where u(n,k,r) = r*binomial((k - 1)*n + r, n)/((k - 1)*n + r).

T(n,k) ~ n*A070914(n,k-2)/(n*(k-2)+2) for fixed k.

A295259 Array read by antidiagonals: T(n,k) = number of nonequivalent dissections of a polygon into n k-gons by nonintersecting diagonals rooted at a cell up to rotation and reflection (k >= 3).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 4, 6, 1, 1, 4, 13, 16, 1, 1, 6, 22, 64, 52, 1, 1, 6, 35, 147, 315, 170, 1, 1, 8, 49, 302, 1074, 1727, 579, 1, 1, 8, 67, 518, 2763, 8216, 9658, 1996, 1, 1, 10, 87, 843, 5916, 27168, 64798, 55657, 7021

Offset: 1

Andrew Howroyd, Nov 18 2017

The polygon prior to dissection will have n*(k-2)+2 sides.

In the Harary, Palmer and Read reference these are the sequences called f.

			Array begins:
  =========================================================
  n\k|    3      4       5        6         7         8
  ---|-----------------------------------------------------
   1 |    1      1       1        1         1         1 ...
   2 |    1      1       1        1         1         1 ...
   3 |    2      4       4        6         6         8 ...
   4 |    6     13      22       35        49        67 ...
   5 |   16     64     147      302       518       843 ...
   6 |   52    315    1074     2763      5916     11235 ...
   7 |  170   1727    8216    27168     70984    159180 ...
   8 |  579   9658   64798   274360    876790   2319678 ...
   9 | 1996  55657  521900  2837208  11069760  34582800 ...
  10 | 7021 325390 4272967 29828330 142148343 524470485 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275
F. Harary, E. M. Palmer and R. C. Read, On the cell-growth problem for arbitrary polygons, Discr. Math. 11 (1975), 371-389.
Wikipedia, Fuss-Catalan number

Columns k=3..5 are A003446, A005035, A005039.

Cf. A033282, A070914, A295222, A295224, A295260.

u[n_, k_, r_] := r*Binomial[(k - 1)*n + r, n]/((k - 1)*n + r);
F[n_, k_] := DivisorSum[GCD[n-1, k], EulerPhi[#]*u[(n-1)/#, k, k/#] &]/k;
T[n_, k_] := (F[n, k] + If[OddQ[k], If[OddQ[n], u[(n-1)/2, k, (k-1)/2], u[n/2-1, k, k-1]], If[OddQ[n], u[(n-1)/2, k, k/2+1], u[n/2-1, k, k]]])/2;
Table[T[n-k-1, k], {n, 1, 14}, {k, n-2, 3, -1}] // Flatten (* Jean-François Alcover, Jan 19 2018, translated from PARI *)

\\ here u is Fuss-Catalan sequence with p = k+1.
u(n,k,r) = {r*binomial((k - 1)*n + r, n)/((k - 1)*n + r)}
F(n,k) = {sumdiv(gcd(n-1,k), d, eulerphi(d)*u((n-1)/d,k,k/d))/k}
T(n,k) = {(F(n,k) + if(k%2, if(n%2, u((n-1)/2,k,(k-1)/2), u(n/2-1,k,(k-1))), if(n%2, u((n-1)/2,k,k/2+1), u(n/2-1,k,k)) ))/2;}
for(n=1, 10, for(k=3, 8, print1(T(n, k), ", ")); print);

from sympy import binomial, gcd, totient, divisors
def u(n, k, r): return r*binomial((k - 1)*n + r, n)//((k - 1)*n + r)
def F(n, k): return sum([totient(d)*u((n - 1)//d, k, k//d) for d in divisors(gcd(n - 1, k))])//k
def T(n, k): return (F(n, k) + ((u((n - 1)//2, k, (k - 1)//2) if n%2 else u(n//2 - 1, k, k - 1)) if k%2 else (u((n - 1)//2, k, k//2 + 1) if n%2 else u(n//2 - 1, k, k))))//2
for n in range(1, 11): print([T(n, k) for k in range(3, 9)]) # Indranil Ghosh, Dec 13 2017, after PARI code

T(n,k) ~ A295222(n,k)/2 for fixed k.

cp's OEIS Frontend

A000207 Number of inequivalent ways of dissecting a regular (n+2)-gon into n triangles by n-1 non-intersecting diagonals under rotations and reflections; also the number of (unlabeled) maximal outerplanar graphs on n+2 vertices.

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A001004 Number of nonequivalent dissections of an (n+2)-gon by nonintersecting diagonals up to rotation and reflection.

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A070914 Array read by antidiagonals giving number of paths up and left from (0,0) to (n,kn) where x/y <= k for all intermediate points.

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A005036 Number of nonequivalent dissections of a polygon into n quadrilaterals by nonintersecting diagonals up to rotation and reflection.

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A295224 Array read by antidiagonals: T(n,k) = number of nonequivalent dissections of a polygon into n k-gons by nonintersecting diagonals up to rotation (k >= 3).

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A004127 Number of planar hexagon trees with n hexagons.

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A295419 Number of dissections of an n-gon by nonintersecting diagonals into polygons with a prime number of sides counted up to rotations and reflections.