A001004 -id:A001004 - cp's OEIS frontend

A002054 Binomial coefficient C(2n+1, n-1).

1, 5, 21, 84, 330, 1287, 5005, 19448, 75582, 293930, 1144066, 4457400, 17383860, 67863915, 265182525, 1037158320, 4059928950, 15905368710, 62359143990, 244662670200, 960566918220, 3773655750150, 14833897694226, 58343356817424, 229591913401900

Offset: 1

N. J. A. Sloane

a(n) = number of permutations in S_{n+2} containing exactly one 312 pattern. E.g., S_3 has a_1 = 1 permutations containing exactly one 312 pattern, and S_4 has a_2 = 5 permutations containing exactly one 312 pattern, namely 1423, 2413, 3124, 3142, and 4231. This comment is also true if 312 is replaced by any of 132, 213, or 231 (but not 123 or 321, for which see A003517). [Comment revised by N. J. A. Sloane, Nov 26 2022]
Number of valleys in all Dyck paths of semilength n+1. Example: a(2)=5 because UD*UD*UD, UD*UUDD, UUDD*UD, UUD*UDD, UUUDDD, where U=(1,1), D=(1,-1) and the valleys are shown by *. - Emeric Deutsch, Dec 05 2003
Number of UU's (double rises) in all Dyck paths of semilength n+1. Example: a(2)=5 because UDUDUD, UDU*UDD, U*UDDUD, U*UDUDD, U*U*UDDD, the double rises being shown by *. - Emeric Deutsch, Dec 05 2003
Number of peaks at level higher than one (high peaks) in all Dyck paths of semilength n+1. Example: a(2)=5 because UDUDUD, UDUU*DD, UU*DDUD, UU*DU*DD, UUU*DDD, the high peaks being shown by *. - Emeric Deutsch, Dec 05 2003
Number of diagonal dissections of a convex (n+3)-gon into n regions. Number of standard tableaux of shape (n,n,1) (see Stanley reference). - Emeric Deutsch, May 20 2004
Number of dissections of a convex (n+3)-gon by noncrossing diagonals into several regions, exactly n-1 of which are triangular. Example: a(2)=5 because the convex pentagon ABCDE is dissected by any of the diagonals AC, BD, CE, DA, EB into regions containing exactly 1 triangle. - Emeric Deutsch, May 31 2004
Number of jumps in all full binary trees with n+1 internal nodes. In the preorder traversal of a full binary tree, any transition from a node at a deeper level to a node on a strictly higher level is called a jump. - Emeric Deutsch, Jan 18 2007
a(n) is the total number of nonempty Dyck subpaths in all Dyck paths (A000108) of semilength n. For example, the Dyck path UUDUUDDD has Dyck subpaths stretching over positions 1-8 (the entire path), 2-3, 2-7, 4-7, 5-6 and so contributes 5 to a(4). - David Callan, Jul 25 2008
a(n+1) is the total number of ascents in the set of all n-permutations avoiding the pattern 132. For example, a(2) = 5 because there are 5 ascents in the set 123, 213, 231, 312, 321. - Cheyne Homberger, Oct 25 2013
Number of increasing tableaux of shape (n+1,n+1) with largest entry 2n+1. An increasing tableau is a semistandard tableau with strictly increasing rows and columns, and set of entries an initial segment of the positive integers. Example: a(2) = 5 counts the five tableaux (124)(235), (123)(245), (124)(345), (134)(245), (123)(245). - Oliver Pechenik, May 02 2014
a(n) is the number of noncrossing partitions of 2n+1 into n-1 blocks of size 2 and 1 block of size 3. - Oliver Pechenik, May 02 2014
Number of paths in the half-plane x>=0, from (0,0) to (2n+1,3), and consisting of steps U=(1,1) and D=(1,-1). For example, for n=2, we have the 5 paths: UUUUD, UUUDU, UUDUU, UDUUU, DUUUU. - José Luis Ramírez Ramírez, Apr 19 2015
From Gus Wiseman, Aug 20 2021: (Start)
Also the number of binary numbers with 2n+2 digits and with two more 0's than 1's. For example, the a(2) = 5 binary numbers are: 100001, 100010, 100100, 101000, 110000, with decimal values 33, 34, 36, 40, 48. Allowing first digit 0 gives A001791, ranked by A345910/A345912.
Also the number of integer compositions of 2n+2 with alternating sum -2, where the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. For example, the a(3) = 21 compositions are:
  (35)  (152)  (1124)  (11141)  (111113)
        (251)  (1223)  (12131)  (111212)
               (1322)  (13121)  (111311)
               (1421)  (14111)  (121112)
               (2114)           (121211)
               (2213)           (131111)
               (2312)
               (2411)
The following pertain to these compositions:
- The unordered version is A344741.
- Ranked by A345924 (reverse: A345923).
- A345197 counts compositions by length and alternating sum.
- A345925 ranks compositions with alternating sum 2 (reverse: A345922).
(End)

			G.f. = x + 5*x^2 + 21*x^3 + 84*x^4 + 330*x^5 + 1287*x^6 + 5005*x^7 + ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
George Grätzer, General Lattice Theory. Birkhauser, Basel, 1998, 2nd edition, p. 474, line -3.
A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992.
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).

G. C. Greubel, Table of n, a(n) for n = 1..1000 (terms 1..100 computed by T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Anwar Al Ghabra, K. Gopala Krishna, Patrick Labelle, and Vasilisa Shramchenko, Enumeration of multi-rooted plane trees, arXiv:2301.09765 [math.CO], 2023.
F. R. Bernhart and N. J. A. Sloane, Emails, April-May 1994.
Jean-Luc Baril and Sergey Kirgizov, The pure descent statistic on permutations, Discrete Mathematics, Vol. 340, No. 10 (2017), pp. 2550-2558; preprint, 2017.
Jean-Luc Baril, Sergey Kirgizov, and Armen Petrossian, Dyck Paths with catastrophes modulo the positions of a given pattern, Australasian J. Comb. (2022) Vol. 84, No. 2, 398-418.
David Callan, A recursive bijective approach to counting permutations containing 3-letter patterns, arXiv:math/0211380 [math.CO], 2002.
Arthur Cayley, On the partitions of a polygon, Proc. London Math. Soc., Vol. 22 (1891), pp. 237-262 = Collected Mathematical Papers, Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 93ff.
Matteo Cervetti and Luca Ferrari, Pattern avoidance in the matching pattern poset, arXiv:2009.01024 [math.CO], 2020.
Colin Defant, Proofs of Conjectures about Pattern-Avoiding Linear Extensions, arXiv:1905.02309 [math.CO], 2019.
Emeric Deutsch, Dyck path enumeration, Discrete Math., Vol. 204, No. 1-3 (1999), pp. 167-202.
Gennady Eremin, Dyck Numbers, IV. Nested patterns in OEIS A036991, arXiv:2306.10318 [math.CO], 2023. See (5) p. 7.
Luca Ferrari and Emanuele Munarini, Enumeration of edges in some lattices of paths, J. Int. Seq., Vol. 17 (2014), Article 14.1.5; arXiv preprint, arXiv:1203.6792 [math.CO], 2012.
Xiaoyu He, Emily Huang, Ihyun Nam and Rishubh Thaper, Shuffle Squares and Reverse Shuffle Squares, arXiv:2109.12455 [math.CO], 2021.
Clemens Heuberger, Sarah J. Selkirk, and Stephan Wagner, Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo k, arXiv:2204.14023 [math.CO], 2022.
Milan Janjic, Two Enumerative Functions.
Werner Krandick, Trees and jumps and real roots, J. Computational and Applied Math., Vol. 162, No. 1 (2004), pp. 51-55.
Toufik Mansour, Statistics on Dyck Paths, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.5.
Toufik Mansour and Alek Vainshtein, Counting occurrences of 123 in a permutation, arXiv:math/0105073 [math.CO], 2001.
Henri Mühle, Symmetric Chain Decompositions and the Strong Sperner Property for Noncrossing Partition Lattices, arXiv:1509.06942 [math.CO], 2015.
Asamoah Nkwanta and Earl R. Barnes, Two Catalan-type Riordan Arrays and their Connections to the Chebyshev Polynomials of the First Kind, Journal of Integer Sequences, Vol. 15 (2012), Article 12.3.3.
John Noonan and Doron Zeilberger, The Enumeration of Permutations With a Prescribed Number of ``Forbidden'' Patterns, arXiv:math/9808080 [math.CO], 1998.
Oliver Pechenik, Cyclic sieving of increasing tableaux and small Schröder paths, arXiv:1209.1355 [math.CO], 2012-2014.
Oliver Pechenik, Cyclic sieving of increasing tableaux and small Schröder paths, J. Combin. Theory A, Vol. 125 (2014), pp. 357-378.
Karol Penson, Hausdorff moment problems for combinatorial numbers: heuristics via Meijer G-functions, Nov. 2022.
Ronald C. Read, On general dissections of a polygon, Aequationes Mathematicae, Vol. 18, No. 1-2 (1978), pp. 370-388; Preprint, 1974.
Mark Shattuck, Enumeration of non-crossing partitions according to subwords with repeated letters, arXiv:2303.06300 [math.CO], 2023.
Richard P. Stanley, Polygon dissections and standard Young tableaux, J. Comb. Theory, Ser. A, Vol. 76 , No. 1 (1996), pp. 175-177.
Daniel W. Stasiuk, An Enumeration Problem for Sequences of n-ary Trees Arising from Algebraic Operads, Master's Thesis, University of Saskatchewan-Saskatoon (2018).
A. Vogt, Resummation of small-x double logarithms in QCD: semi-inclusive electron-positron annihilation, arXiv:1108.2993 [hep-ph], 2011.
N. J. Wildberger and Dean Rubine, A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode, Amer. Math. Monthly (2025). See section 12.

Diagonal 4 of triangle A100257. Also a diagonal of A033282.
Equals (1/2) A024483(n+2). Bisection of A037951 and A037955.
Cf. A001263.
Column k=1 of A263771.
Counts terms of A031445 with 2n+2 digits in binary.
Cf. A000097, A000346, A000984, A001622, A001700, A007318, A008549, A031444, A058622, A097805, A116406, A138364, A163493, A202736.
Cf. binomial(2*n+m, n): A000984 (m = 0), A001700 (m = 1), A001791 (m = 2), A002694 (m = 4), A003516 (m = 5), A002696 (m = 6), A030053 - A030056, A004310 - A004318.

List([1..25],n->Binomial(2*n+1,n-1)); # Muniru A Asiru, Aug 09 2018

[Binomial(2*n+1, n-1): n in [1..30]]; // Vincenzo Librandi, Apr 20 2015

with(combstruct): seq((count(Composition(2*n+2), size=n)), n=1..24); # Zerinvary Lajos, May 03 2007

CoefficientList[Series[8/(((Sqrt[1-4x] +1)^3)*Sqrt[1-4x]), {x,0,22}], x] (* Robert G. Wilson v, Aug 08 2011 *)
a[ n_]:= Binomial[2 n + 1, n - 1]; (* Michael Somos, Apr 25 2014 *)

PARI
```
{a(n) = binomial( 2*n+1, n-1)};
```

from _future_ import division
A002054_list, b = [], 1
for n in range(1,10**3):
    A002054_list.append(b)
    b = b*(2*n+2)*(2*n+3)//(n*(n+3)) # Chai Wah Wu, Jan 26 2016

[binomial(2*n+1, n-1) for n in (1..25)] # G. C. Greubel, Mar 22 2019

a(n) = Sum_{j=0..n-1} binomial(2*j, j) * binomial(2*n - 2*j, n-j-1)/(j+1). - Yong Kong (ykong(AT)curagen.com), Dec 26 2000
G.f.: z*C^4/(2-C), where C=[1-sqrt(1-4z)]/(2z) is the Catalan function. - Emeric Deutsch, Jul 05 2003
From Wolfdieter Lang, Jan 09 2004: (Start)
a(n) = binomial(2*n+1, n-1) = n*C(n+1)/2, C(n)=A000108(n) (Catalan).
G.f.: (1 - 2*x - (1-3*x)*c(x))/(x*(1-4*x)) with g.f. c(x) of A000108. (End)
G.f.: z*C(z)^3/(1-2*z*C(z)), where C(z) is the g.f. of Catalan numbers. - José Luis Ramírez Ramírez, Apr 19 2015
G.f.: 2F1(5/2, 2; 4; 4*x). - R. J. Mathar, Aug 09 2015
D-finite with recurrence: a(n+1) = a(n)*(2*n+3)*(2*n+2)/(n*(n+3)). - Chai Wah Wu, Jan 26 2016
From Ilya Gutkovskiy, Aug 30 2016: (Start)
E.g.f.: (BesselI(0,2*x) + (1 - 1/x)*BesselI(1,2*x))*exp(2*x).
a(n) ~ 2^(2*n+1)/sqrt(Pi*n). (End)
a(n) = (1/(n+1))*Sum_{i=0..n-1} (n+1-i)*binomial(2n+2,i), n >= 1. - Taras Goy, Aug 09 2018
G.f.: (x - 1 + (1 - 3*x)/sqrt(1 - 4*x))/(2*x^2). - Michael Somos, Jul 28 2021
From Amiram Eldar, Jan 24 2022: (Start)
Sum_{n>=1} 1/a(n) = 5/3 - 2*Pi/(9*sqrt(3)).
Sum_{n>=1} (-1)^(n+1)/a(n) = 52*log(phi)/(5*sqrt(5)) - 7/5, where phi is the golden ratio (A001622). (End)
a(n) = A001405(2*n+1) - A000108(n+1), n >= 1 (from Eremin link, page 7). - Gennady Eremin, Sep 05 2023
G.f.: x/(1 - 4*x)^2 * c(-x/(1 - 4*x))^3, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. - Peter Bala, Feb 03 2024
From Peter Bala, Oct 13 2024: (Start)
a(n) = Integral_{x = 0..4} x^n * w(x) dx, where the weight function w(x) = 1/(2*Pi) * sqrt(x)*(x - 3)/sqrt(4 - x) (see Penson).
G.f. x*/sqrt(1 - 4*x) * c(x)^3. (End)

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).

Original entry on oeis.org

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

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]))

A002055 Number of diagonal dissections of a convex n-gon into n-4 regions.

Original entry on oeis.org

1, 9, 56, 300, 1485, 7007, 32032, 143208, 629850, 2735810, 11767536, 50220040, 212952285, 898198875, 3771484800, 15775723920, 65770848990, 273420862110, 1133802618000, 4691140763400, 19371432850770, 79850555673174

Offset: 5

N. J. A. Sloane

Number of standard tableaux of shape (n-4,n-4,1,1) (see Stanley reference). - Emeric Deutsch, May 20 2004
Number of increasing tableaux of shape (n-2,n-2) with largest entry 2n-6. An increasing tableau is a semistandard tableau with strictly increasing rows and columns, and set of entries an initial segment of the positive integers. - Oliver Pechenik, May 02 2014
a(n) = number of noncrossing partitions of 2n-6 into n-4 blocks, each of size at least 2. - Oliver Pechenik, May 02 2014

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 = 5..100
D. Beckwith, Legendre polynomials and polygon dissections?, Amer. Math. Monthly, 105 (1998), 256-257.
A. Cayley, On the partitions of a polygon, Proc. London Math. Soc., 22 (1891), 237-262 = Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 93ff.
P. Lisonek, Closed forms for the number of polygon dissections, Journal of Symbolic Computation 20 (1995), 595-601.
O. Pechenik, Cyclic sieving of increasing tableaux and small Schröder paths, arXiv:1209.1355 [math.CO], 2012-2014.
O. Pechenik, Cyclic sieving of increasing tableaux and small Schröder paths, J. Combin. Theory A, 125 (2014), 357-378.
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, Table 1.
R. P. Stanley, Polygon dissections and standard Young tableaux, J. Comb. Theory, Ser. A, 76, 175-177, 1996.

a(n) = f(n,n+1) where f is given in A034261.

Table[(Binomial[n-3,2]Binomial[2n-6,n-5])/(n-4),{n,5,30}] (* Harvey P. Dale, Nov 06 2011 *)

a(n) = (binomial(n - 3, 2) * binomial(2*n - 6, n - 5))/(n - 4);
for(n=5, 30, print1(a(n),", ")) \\ Indranil Ghosh, Apr 11 2017

a(n) = binomial(n-3, 2)*binomial(2*n-6, n-5)/(n-4).
With offset 0, this has a(n)=(n+2)*C(2n+4,n)/2 and e.g.f. dif(dif(x*dif(exp(2x)*Bessel_I(2,2x),x),x),x)/2. - Paul Barry, Aug 25 2007
G.f.: 16*x^5*(x+sqrt(1-4x))/((1-4x)^(3/2) *(1+sqrt(1-4x))^4 ). - R. J. Mathar, Nov 17 2011
D-finite with recurrence: (n-1)*a(n) +(23-11n)*a(n-1) +10*(4n-13)*a(n-2) +10*(23-5n)*a(n-3) +4*(2n-13)*a(n-4)=0. - R. J. Mathar, Nov 17 2011
a(n) ~ 4^n*sqrt(n)/(128*sqrt(Pi)). - Ilya Gutkovskiy, Apr 11 2017

A295634 Triangle read by rows: T(n,k) = number of nonequivalent dissections of an n-gon into k polygons by nonintersecting diagonals up to rotation and reflection.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 3, 3, 1, 2, 6, 7, 4, 1, 3, 11, 24, 24, 12, 1, 3, 17, 51, 89, 74, 27, 1, 4, 26, 109, 265, 371, 259, 82, 1, 4, 36, 194, 660, 1291, 1478, 891, 228, 1, 5, 50, 345, 1477, 3891, 6249, 6044, 3176, 733, 1, 5, 65, 550, 3000, 10061, 21524, 29133, 24302, 11326, 2282

Offset: 3

Andrew Howroyd, Nov 24 2017

			Triangle begins: (n >= 3, k >= 1)
1;
1, 1;
1, 1,  1;
1, 2,  3,   3;
1, 2,  6,   7,   4;
1, 3, 11,  24,  24,   12;
1, 3, 17,  51,  89,   74,   27;
1, 4, 26, 109, 265,  371,  259,  82;
1, 4, 36, 194, 660, 1291, 1478, 891, 228;
...

Andrew Howroyd, Table of n, a(n) for n = 3..1277

Row sums are A001004.
Column k=3 is A003453.
Diagonals include A000207, A003449, A003450.
Cf. A295260, A295419, A295633.

\\ See A295419 for DissectionsModDihedral()
T=DissectionsModDihedral(apply(i->y, [1..12]));
for(n=3, #T, for(k=1, n-2, print1(polcoeff(T[n], k), ", ")); print)

A003454 Number of nonequivalent dissections of an n-gon by nonintersecting diagonals rooted at a cell up to rotation.

Original entry on oeis.org

1, 2, 6, 25, 107, 509, 2468, 12258, 61797, 315830, 1630770, 8498303, 44629855, 235974495, 1255105304, 6710883952, 36050676617, 194478962422, 1053120661726, 5722375202661, 31191334491891, 170504130213135, 934495666529380, 5134182220623958, 28270742653671621

Offset: 3

N. J. A. Sloane

nonn

Total number of dissections of an n-gon into polygons without reflection and rooted at a cell. - Sean A. Irvine, May 14 2015

Say two n-gons are equivalent (or in the same convexity class) if there is a bijection between the edges and vertices which preserves inclusion of vertices and edges, preserves the handedness of the polygon (does not reflect the polygon over a line), maps vertices of the convex hulls to each other, and induces an equivalence on each nontrivially connected component of Hull(X) \ X. This sequence is the number of convexity classes for an n-gon, up to rotation. - Griffin N. Macris, Mar 02 2021

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 = 3..200
P. Lisonek, Closed forms for the number of polygon dissections, Journal of Symbolic Computation 20 (1995), 595-601.
Ronald C. Read, On general dissections of a polygon, Aequat. math. 18 (1978) 370-388.

Cf. A001004, A003455, A003456, A005033, A295222.

Cf. A003442.

\\ See A003442 for DissectionsModCyclicRooted.
DissectionsModCyclicRooted(apply(i->1, [1..30])) \\ Andrew Howroyd, Nov 22 2017

G.f.: -f(x) - (f(x)^2 + f(x^2))/2 + Sum_{k>=1} (phi(k)/k)*log(1/(1 - f(x^k))), where phi(k) is Euler's Totient function and f(x) = (1 + x - sqrt(1 - 6x + x^2))/4 is related to the o.g.f. for A001003. - Griffin N. Macris, Mar 02 2021

More terms from Sean A. Irvine, May 14 2015

Name clarified by Andrew Howroyd, Nov 22 2017

A003455 Number of nonequivalent dissections of an n-gon by nonintersecting diagonals up to rotation.

Original entry on oeis.org

1, 2, 3, 11, 29, 122, 479, 2113, 9369, 43392, 203595, 975563, 4736005, 23296394, 115811855, 581324861, 2942579633, 15008044522, 77064865555, 398150807179, 2068470765261, 10800665952376, 56658467018647, 298489772155137, 1578702640556193

Offset: 3

N. J. A. Sloane

nonn

Total number of dissections of an n-gon into polygons without reflection. - Sean A. Irvine, May 15 2015

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 = 3..200
P. Lisonek, Closed forms for the number of polygon dissections, Journal of Symbolic Computation 20 (1995), 595-601.
Ronald C. Read, On general dissections of a polygon, Aequat. math. 18 (1978) 370-388.

Cf. A001004, A003454, A003456, A005034, A295224, A295495.

\\ See A295495 for DissectionsModCyclic().
DissectionsModCyclic(apply(v->1, [1..30])) \\ Andrew Howroyd, Nov 22 2017

More terms from Sean A. Irvine, May 15 2015

Name clarified by Andrew Howroyd, Nov 22 2017

A002056 Number of diagonal dissections of a convex n-gon into n-5 regions.

Original entry on oeis.org

1, 14, 120, 825, 5005, 28028, 148512, 755820, 3730650, 17978180, 84987760, 395482815, 1816357725, 8250123000, 37119350400, 165645101160, 733919156190, 3231337461300, 14147884842000, 61636377252450, 267325773340626, 1154761882042824, 4969989654817600

Offset: 6

N. J. A. Sloane

nonn

Number of standard tableaux of shape (n-5,n-5,1,1,1) (see Stanley reference). - Emeric Deutsch, May 20 2004
Number of increasing tableaux of shape (n-2,n-2) with largest entry 2n-7. An increasing tableau is a semistandard tableau with strictly increasing rows and columns, and set of entries an initial segment of the positive integers. - Oliver Pechenik, May 02 2014
Number of noncrossing partitions of 2n-7 into n-5 blocks all of size at least 2. - Oliver Pechenik, May 02 2014

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=6..100
D. Beckwith, Legendre polynomials and polygon dissections?, Amer. Math. Monthly, 105 (1998), 256-257.
A. Cayley, On the partitions of a polygon, Proc. London Math. Soc., 22 (1891), 237-262 = Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 93ff.
P. Lisonek, Closed forms for the number of polygon dissections, Journal of Symbolic Computation 20 (1995), 595-601.
O. Pechenik, Cyclic sieving of increasing tableaux and small Schröder paths, arXiv:1209.1355 [math.CO], 2012-2014.
O. Pechenik, Cyclic sieving of increasing tableaux and small Schröder paths, J. Combin. Theory A, 125 (2014), 357-378.
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.
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, Table 1.
R. P. Stanley, Polygon dissections and standard Young tableaux, J. Comb. Theory, Ser. A, 76, 175-177, 1996.

[Binomial(n-3, 3)*Binomial(2*n-7, n-6)/(n-5): n in [6..30]]; // Vincenzo Librandi, Feb 18 2020

A002056:=n->binomial(n-3,3)*binomial(2*n-7,n-6)/(n-5): seq(A002056(n), n=6..40); # Wesley Ivan Hurt, Apr 12 2017

Table[Binomial[n - 3, 3] Binomial[2n - 7, n - 6]/(n - 5), {n, 6, 50}] (* Indranil Ghosh, Apr 11 2017 *)

a(n) = binomial(n-3, 3)*binomial(2n-7, n-6)/(n-5).
G.f.: (x-1+(1-11*x+40*x^2-50*x^3+10*x^4)*(1-4*x)^(-5/2))/(2*x^5). - Mark van Hoeij, Oct 25 2011
a(n) ~ 4^n*n^(3/2)/(768*sqrt(Pi)). - Ilya Gutkovskiy, Apr 11 2017
D-finite with recurrence: -(n-1)*(n-5)*(n-6)*a(n) +2*(2*n-7)*(n-3)*(n-4)*a(n-1)=0. - R. J. Mathar, Feb 16 2020

A003456 Number of nonequivalent dissections of an n-gon by nonintersecting diagonals rooted at a cell up to rotation and reflection.

Original entry on oeis.org

1, 2, 5, 17, 62, 275, 1272, 6225, 31075, 158376, 816229, 4251412, 22319056, 117998524, 627573216, 3355499036, 18025442261, 97239773408, 526560862829, 2861189112867, 15595669996482, 85252072993968, 467247847612316, 2567091151780343

Offset: 3

N. J. A. Sloane

nonn

Total number of dissections of an n-gon into polygons with reflection and rooted at a cell. - Sean A. Irvine, May 14 2015

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 = 3..200
P. Lisonek, Closed forms for the number of polygon dissections, Journal of Symbolic Computation 20 (1995), 595-601.
Ronald C. Read, On general dissections of a polygon, Aequat. math. 18 (1978) 370-388.

Cf. A001004, A003454, A003455, A005035, A295259.

\\ See A003447 for DissectionsModDihedralRooted()
DissectionsModDihedralRooted(apply(i->1, [1..30]))

More terms from Sean A. Irvine, May 14 2015
Name clarified by Andrew Howroyd, Nov 24 2017
a(15) corrected by Andrew Howroyd, Nov 24 2017

A003441 Number of nonequivalent dissections of a polygon into n triangles by nonintersecting diagonals rooted at a cell up to rotation.

Original entry on oeis.org

1, 1, 3, 10, 30, 99, 335, 1144, 3978, 14000, 49742, 178296, 643856, 2340135, 8554275, 31429068, 115997970, 429874830, 1598952498, 5967382200, 22338765540, 83859016956, 315614844558, 1190680751376, 4501802224520, 17055399281284

Offset: 1

N. J. A. Sloane

nonn

Number of dissections of regular (n+2)-gon into n polygons without reflection and rooted at a cell. - Sean A. Irvine, May 05 2015

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 = 1..200
Paul Drube and Puttipong Pongtanapaisan, Annular Non-Crossing Matchings, Journal of Integer Sequences, Vol. 19 (2016), #16.2.4.
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.
P. Lisonek, Closed forms for the number of polygon dissections, Journal of Symbolic Computation 20 (1995), 595-601.
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.

Column k=3 of A295222.

[seq(combstruct[count]([C, {C=Cycle(BT,card=3),BT=Union(Z,Prod(BT,BT))}],size=n),n=0..12)];

a[n_] := DivisorSum[GCD[3, n-1], EulerPhi[#] Binomial[(2n+1)/#, (n-1)/#]/ (2n+1)&];
Array[a, 30] (* Jean-François Alcover, Jul 02 2018 *)

catalan(n) = binomial(2*n, n)/(n+1);
a(n) = binomial(2*n+1, n-1)/(2*n+1) + 2/3*(if ((n-1) % 3, 0, catalan((n-1)/3))); \\ Michel Marcus, Jan 23 2016

a(n) = number of necklaces of n-1 white beads and n+2 black beads. a(n) = binomial(2n+1, n-1)/(2n+1) + (2/3)*C((n-1)/3) where C is the Catalan number A000108 (assumed to be 0 for nonintegral argument). G.f.: ( ((1-sqrt(1-4x))/2)^3 + (1-sqrt(1-4x^3)) )/(3x^2).

Numbers so far suggest that two trisections of sequence agree with those of A050181. - Ralf Stephan, Mar 28 2004

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003

Name edited by Andrew Howroyd, Nov 20 2017

cp's OEIS Frontend

A002054 Binomial coefficient C(2n+1, n-1).

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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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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.

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A002055 Number of diagonal dissections of a convex n-gon into n-4 regions.

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A295634 Triangle read by rows: T(n,k) = number of nonequivalent dissections of an n-gon into k polygons by nonintersecting diagonals up to rotation and reflection.

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A003454 Number of nonequivalent dissections of an n-gon by nonintersecting diagonals rooted at a cell up to rotation.

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A003455 Number of nonequivalent dissections of an n-gon by nonintersecting diagonals up to rotation.

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