A047749 -id:A047749 - cp's OEIS frontend

A001764 a(n) = binomial(3n,n)/(2n+1) (enumerates ternary trees and also noncrossing trees).

1, 1, 3, 12, 55, 273, 1428, 7752, 43263, 246675, 1430715, 8414640, 50067108, 300830572, 1822766520, 11124755664, 68328754959, 422030545335, 2619631042665, 16332922290300, 102240109897695, 642312451217745, 4048514844039120, 25594403741131680, 162250238001816900

Offset: 0

N. J. A. Sloane

Smallest number of straight line crossing-free spanning trees on n points in the plane.
Number of dissections of some convex polygon by nonintersecting diagonals into polygons with an odd number of sides and having a total number of 2n+1 edges (sides and diagonals). - Emeric Deutsch, Mar 06 2002
Number of lattice paths of n East steps and 2n North steps from (0,0) to (n,2n) and lying weakly below the line y=2x. - David Callan, Mar 14 2004
With interpolated zeros, this has g.f. 2*sqrt(3)*sin(arcsin(3*sqrt(3)*x/2)/3)/(3*x) and a(n) = C(n+floor(n/2),floor(n/2))*C(floor(n/2),n-floor(n/2))/(n+1). This is the first column of the inverse of the Riordan array (1-x^2,x(1-x^2)) (essentially reversion of y-y^3). - Paul Barry, Feb 02 2005
Number of 12312-avoiding matchings on [2n].
Number of complete ternary trees with n internal nodes, or 3n edges.
Number of rooted plane trees with 2n edges, where every vertex has even outdegree ("even trees").
a(n) is the number of noncrossing partitions of [2n] with all blocks of even size. E.g.: a(2)=3 counts 12-34, 14-23, 1234. - David Callan, Mar 30 2007
Pfaff-Fuss-Catalan sequence C^{m}_n for m=3, see the Graham et al. reference, p. 347. eq. 7.66.
Also 3-Raney sequence, see the Graham et al. reference, p. 346-7.
The number of lattice paths from (0,0) to (2n,0) using an Up-step=(1,1) and a Down-step=(0,-2) and staying above the x-axis. E.g., a(2) = 3; UUUUDD, UUUDUD, UUDUUD. - Charles Moore (chamoore(AT)howard.edu), Jan 09 2008
a(n) is (conjecturally) the number of permutations of [n+1] that avoid the patterns 4-2-3-1 and 4-2-5-1-3 and end with an ascent. For example, a(4)=55 counts all 60 permutations of [5] that end with an ascent except 42315, 52314, 52413, 53412, all of which contain a 4-2-3-1 pattern and 42513. - David Callan, Jul 22 2008
Central terms of pendular triangle A167763. - Philippe Deléham, Nov 12 2009
With B(x,t)=x+t*x^3, the comp. inverse in x about 0 is A(x,t) = Sum_{j>=0} a(j) (-t)^j x^(2j+1). Let U(x,t)=(x-A(x,t))/t. Then DU(x,t)/Dt=dU/dt+U*dU/dx=0 and U(x,0)=x^3, i.e., U is a solution of the inviscid Burgers's, or Hopf, equation. Also U(x,t)=U(x-t*U(x,t),0) and dB(x,t)/dt = U(B(x,t),t) = x^3 = U(x,0). The characteristics for the Hopf equation are x(t) = x(0) + t*U(x(t),t) = x(0) + t*U(x(0),0) = x(0) + t*x(0)^3 = B(x(0),t). These results apply to all the Fuss-Catalan sequences with 3 replaced by n>0 and 2 by n-1 (e.g., A000108 with n=2 and A002293 with n=4), see also A086810, which can be generalized to A133437, for associahedra. - Tom Copeland, Feb 15 2014
Number of intervals (i.e., ordered pairs (x,y) such that x<=y) in the Kreweras lattice (noncrossing partitions ordered by refinement) of size n, see the Bernardi & Bonichon (2009) and Kreweras (1972) references. - Noam Zeilberger, Jun 01 2016
Number of sum-indecomposable (4231,42513)-avoiding permutations. Conjecturally, number of sum-indecomposable (2431,45231)-avoiding permutations. - Alexander Burstein, Oct 19 2017
a(n) is the number of topologically distinct endstates for the game Planted Brussels Sprouts on n vertices, see Ji and Propp link. - Caleb Ji, May 14 2018
Number of complete quadrillages of 2n+2-gons. See Baryshnikov p. 12. See also Nov 10 2014 comments in A134264. - Tom Copeland, Jun 04 2018
a(n) is the number of 2-regular words on the alphabet [n] that avoid the patterns 231 and 221. Equivalently, this is the number of 2-regular tortoise-sortable words on the alphabet [n] (see the Defant and Kravitz link). - Colin Defant, Sep 26 2018
a(n) is the number of Motzkin paths of length 3n with n steps of each type, with the condition that (1, 0) and (1, 1) steps alternate (starting with (1, 0)). - Helmut Prodinger, Apr 08 2019
a(n) is the number of uniquely sorted permutations of length 2n+1 that avoid the patterns 312 and 1342. - Colin Defant, Jun 08 2019
The compositional inverse o.g.f. pair in Copeland's comment above are related to a pair of quantum fields in Balduf's thesis by Theorem 4.2 on p. 92. - Tom Copeland, Dec 13 2019
The sequences of Fuss-Catalan numbers, of which this is the first after the Catalan numbers A000108 (the next is A002293), appear in articles on random matrices and quantum physics. See Banica et al., Collins et al., and Mlotkowski et al. Interpretations of these sequences in terms of the cardinality of specific sets of noncrossing partitions are provided by A134264. - Tom Copeland, Dec 21 2019
Call C(p, [alpha], g) the number of partitions of a cyclically ordered set with p elements, of cyclic type [alpha], and of genus g (the genus g Faa di Bruno coefficients of type [alpha]). This sequence counts the genus 0 partitions (non-crossing, or planar, partitions) of p = 3n into n parts of length 3: a(n) = C(3n, [3^n], 0). For genus 1 see A371250, for genus 2 see A371251. - Robert Coquereaux, Mar 16 2024
a(n) is the total number of down steps before the first up step in all 2_1-Dyck paths of length 3*n for n > 0. A 2_1-Dyck path is a lattice path with steps (1,2), (1,-1) that starts and ends at y = 0 and does not go below the line y = -1. - Sarah Selkirk, May 10 2020
a(n) is the number of pairs (A<=B) of noncrossing partitions of [n]. - Francesca Aicardi, May 28 2022
a(n) is the number of parking functions of size n avoiding the patterns 231 and 321. - Lara Pudwell, Apr 10 2023
Number of rooted polyominoes composed of n square cells of the hyperbolic regular tiling with Schläfli symbol {4,oo}. A rooted polyomino has one external edge identified, and chiral pairs are counted as two. A stereographic projection of the {4,oo} tiling on the Poincaré disk can be obtained via the Christensson link. - Robert A. Russell, Jan 27 2024
This is instance k = 3 of the family {C(k, n)}A130564.%20-%20_Wolfdieter%20Lang">{n>=0} given in a comment in A130564. - _Wolfdieter Lang, Feb 05 2024
The number of Apollonian networks (planar 3-trees) with n+3 vertices with a given base triangle. - Allan Bickle, Feb 20 2024
Number of rooted polyominoes composed of n tetrahedral cells of the hyperbolic regular tiling with Schläfli symbol {3,3,oo}. A rooted polyomino has one external face identified, and chiral pairs are counted as two. a(n) = T(n) in the second Beineke and Pippert link. - Robert A. Russell, Mar 20 2024

			a(2) = 3 because the only dissections with 5 edges are given by a square dissected by any of the two diagonals and the pentagon with no dissecting diagonal.
G.f. = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 + 7752*x^7 + 43263*x^8 + ...

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Index entries for "core" sequences
Index entries for sequences related to trees

Cf. A001762, A001763, A002294 - A002296, A006013, A025174, A063548, A064017, A072247, A072248, A134264, A143603, A258708, A256311, A188687 (binomial transform), A346628 (inverse binomial transform).
A column of triangle A102537.
Bisection of A047749 and A047761.
Row sums of triangles A108410 and A108767.
Second column of triangle A062993.
Mod 3 = A113047.
2D Polyominoes: A005034 (oriented), A005036 (unoriented), A369315 (chiral), A047749 (achiral), A000108 {3,oo}, A002293 {5,oo}.
3D Polyominoes: A007173 (oriented), A027610 (unoriented), A371350 (chiral), A371351 (achiral).
Cf. A130564 (for C(k, n) cases).

List([0..25],n->Binomial(3*n,n)/(2*n+1)); # Muniru A Asiru, Oct 31 2018

a001764 n = a001764_list !! n
a001764_list = 1 : [a258708 (2 * n) n | n <- [1..]]
-- Reinhard Zumkeller, Jun 23 2015

[Binomial(3*n,n)/(2*n+1): n in [0..30]]; // Vincenzo Librandi, Sep 04 2014

A001764 := n->binomial(3*n,n)/(2*n+1): seq(A001764(n), n=0..25);
with(combstruct): BB:=[T,{T=Prod(Z,F),F=Sequence(B),B=Prod(F,Z,F)}, unlabeled]:seq(count(BB,size=i),i=0..22); # Zerinvary Lajos, Apr 22 2007
with(combstruct):BB:=[S, {B = Prod(S,S,Z), S = Sequence(B)}, labelled]: seq(count(BB, size=n)/n!, n=0..21); # Zerinvary Lajos, Apr 25 2008
n:=30:G:=series(RootOf(g = 1+x*g^3, g),x=0,n+1):seq(coeff(G,x,k),k=0..n); # Robert FERREOL, Apr 03 2015
alias(PS=ListTools:-PartialSums): A001764List := proc(m) local A, P, n;
A := [1,1]; P := [1]; for n from 1 to m - 2 do P := PS(PS([op(P), P[-1]]));
A := [op(A), P[-1]] od; A end: A001764List(25); # Peter Luschny, Mar 26 2022

InverseSeries[Series[y-y^3, {y, 0, 24}], x] (* then a(n)=y(2n+1)=ways to place non-crossing diagonals in convex (2n+4)-gon so as to create only quadrilateral tiles *) (* Len Smiley, Apr 08 2000 *)
Table[Binomial[3n,n]/(2n+1),{n,0,25}] (* Harvey P. Dale, Jul 24 2011 *)

{a(n) = if( n<0, 0, (3*n)! / n! / (2*n + 1)!)};

{a(n) = if( n<0, 0, polcoeff( serreverse( x - x^3 + O(x^(2*n + 2))), 2*n + 1))};

{a(n) = my(A); if( n<0, 0, A = 1 + O(x); for( m=1, n, A = 1 + x * A^3); polcoeff(A, n))};

b=vector(22);b[1]=1;for(n=2,22,for(i=1,n-1,for(j=1,n-1,for(k=1,n-1,if((i-1)+(j-1)+(k-1)-(n-2),NULL,b[n]=b[n]+b[i]*b[j]*b[k])))));a(n)=b[n+1]; print1(a(0));for(n=1,21,print1(", ",a(n))) \\ Gerald McGarvey, Oct 08 2008

Vec(1 + serreverse(x / (1+x)^3 + O(x^30))) \\ Gheorghe Coserea, Aug 05 2015

from math import comb
def A001764(n): return comb(3*n,n)//(2*n+1) # Chai Wah Wu, Nov 10 2022

def A001764_list(n) :
    D = [0]*(n+1); D[1] = 1
    R = []; b = false; h = 1
    for i in range(2*n) :
        for k in (1..h) : D[k] += D[k-1]
        if not b : R.append(D[h])
        else : h += 1
        b = not b
    return R
A001764_list(22) # Peter Luschny, May 03 2012

From Karol A. Penson, Nov 08 2001: (Start)
G.f.: (2/sqrt(3*x))*sin((1/3)*arcsin(sqrt(27*x/4))).
E.g.f.: hypergeom([1/3, 2/3], [1, 3/2], 27/4*x).
Integral representation as n-th moment of a positive function on [0, 27/4]: a(n) = Integral_{x=0..27/4} (x^n*((1/12) * 3^(1/2) * 2^(1/3) * (2^(1/3)*(27 + 3 * sqrt(81 - 12*x))^(2/3) - 6 * x^(1/3))/(Pi * x^(2/3)*(27 + 3 * sqrt(81 - 12*x))^(1/3)))), n >= 0. This representation is unique. (End)
G.f. A(x) satisfies A(x) = 1+x*A(x)^3 = 1/(1-x*A(x)^2) [Cyvin (1998)]. - Ralf Stephan, Jun 30 2003
a(n) = n-th coefficient in expansion of power series P(n), where P(0) = 1, P(k+1) = 1/(1 - x*P(k)^2).
G.f. Rev(x/c(x))/x, where c(x) is the g.f. of A000108 (Rev=reversion of). - Paul Barry, Mar 26 2010
From Gary W. Adamson, Jul 07 2011: (Start)
Let M = the production matrix:
  1, 1
  2, 2, 1
  3, 3, 2, 1
  4, 4, 3, 2, 1
  5, 5, 4, 3, 2, 1
  ...
a(n) = upper left term in M^n. Top row terms of M^n = (n+1)-th row of triangle A143603, with top row sums generating A006013: (1, 2, 7, 30, 143, 728, ...). (End)
Recurrence: a(0)=1; a(n) = Sum_{i=0..n-1, j=0..n-1-i} a(i)a(j)a(n-1-i-j) for n >= 1 (counts ternary trees by subtrees of the root). - David Callan, Nov 21 2011
G.f.: 1 + 6*x/(Q(0) - 6*x); Q(k) = 3*x*(3*k + 1)*(3*k + 2) + 2*(2*(k^2) + 5*k +3) - 6*x*(2*(k^2) + 5*k + 3)*(3*k + 4)*(3*k + 5)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 27 2011
D-finite with recurrence: 2*n*(2n+1)*a(n) - 3*(3n-1)*(3n-2)*a(n-1) = 0. - R. J. Mathar, Dec 14 2011
REVERT transform of A115140. BINOMIAL transform is A188687. SUMADJ transform of A188678. HANKEL transform is A051255. INVERT transform of A023053. INVERT transform is A098746. - Michael Somos, Apr 07 2012
(n + 1) * a(n) = A174687(n).
G.f.: F([2/3,4/3], [3/2], 27/4*x) / F([2/3,1/3], [1/2], (27/4)*x) where F() is the hypergeometric function. - Joerg Arndt, Sep 01 2012
a(n) = binomial(3*n+1, n)/(3*n+1) = A062993(n+1,1). - Robert FERREOL, Apr 03 2015
a(n) = A258708(2*n,n) for n > 0. - Reinhard Zumkeller, Jun 23 2015
0 = a(n)*(-3188646*a(n+2) + 20312856*a(n+3) - 11379609*a(n+4) + 1437501*a(n+5)) + a(n+1)*(177147*a(n+2) - 2247831*a(n+3) + 1638648*a(n+4) - 238604*a(n+5)) + a(n+2)*(243*a(n+2) + 31497*a(n+3) - 43732*a(n+4) + 8288*a(n+5)) for all integer n. - Michael Somos, Jun 03 2016
a(n) ~ 3^(3*n + 1/2)/(sqrt(Pi)*4^(n+1)*n^(3/2)). - Ilya Gutkovskiy, Nov 21 2016
Given g.f. A(x), then A(1/8) = -1 + sqrt(5), A(2/27) = (-1 + sqrt(3))*3/2, A(4/27) = 3/2, A(3/64) = -2 + 2*sqrt(7/3), A(5/64) = (-1 + sqrt(5))*2/sqrt(5), etc. A(n^2/(n+1)^3) = (n+1)/n if n > 1. - Michael Somos, Jul 17 2018
From Peter Bala, Sep 14 2021: (Start)
A(x) = exp( Sum_{n >= 1} (1/3)*binomial(3*n,n)*x^n/n ).
The sequence defined by b(n) := [x^n] A(x)^n = A224274(n) for n >= 1 and satisfies the congruence b(p) == b(1) (mod p^3) for prime p >= 3. Cf. A060941. (End)
G.f.: 1/sqrt(B(x)+(1-6*x)/(9*B(x))+1/3), with B(x):=((27*x^2-18*x+2)/54-(x*sqrt((-(4-27*x))*x))/(2*3^(3/2)))^(1/3). - Vladimir Kruchinin, Sep 28 2021
x*A'(x)/A(x) = (A(x) - 1)/(- 2*A(x) + 3) = x + 5*x^2 + 28*x^3 + 165*x^4 + ... is the o.g.f. of A025174. Cf. A002293 - A002296. - Peter Bala, Feb 04 2022
a(n) = hypergeom([1 - n, -2*n], [2], 1). Row sums of A108767. - Peter Bala, Aug 30 2023
G.f.: z*exp(3*z*hypergeom([1, 1, 4/3, 5/3], [3/2, 2, 2], (27*z)/4)) + 1.
 - Karol A. Penson, Dec 19 2023
G.f.: hypergeometric([1/3, 2/3], [3/2], (3^3/2^2)*x). See the e.g.f. above. - Wolfdieter Lang, Feb 04 2024
a(n) = (3*n)! / (n!*(2*n+1)!). - Allan Bickle, Feb 20 2024
Sum_{n >= 0} a(n)*x^n/(1 + x)^(3*n+1) = 1. See A316371 and A346627. - Peter Bala, Jun 02 2024
G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^5). - Seiichi Manyama, Jun 16 2025

A006013 a(n) = binomial(3*n+1,n)/(n+1).

Original entry on oeis.org

1, 2, 7, 30, 143, 728, 3876, 21318, 120175, 690690, 4032015, 23841480, 142498692, 859515920, 5225264024, 31983672534, 196947587823, 1219199353190, 7583142491925, 47365474641870, 296983176369495, 1868545312633440, 11793499763070480

Offset: 0

N. J. A. Sloane

Enumerates pairs of ternary trees [Knuth, 2014]. - N. J. A. Sloane, Dec 09 2014
G.f. (offset 1) is series reversion of x - 2x^2 + x^3.
Hankel transform is A005156(n+1). - Paul Barry, Jan 20 2007
a(n) = number of ways to connect 2*n - 2 points labeled 1, 2, ..., 2*n-2 in a line with 0 or more noncrossing arcs above the line such that each maximal contiguous sequence of isolated points has even length. For example, with arcs separated by dashes, a(3) = 7 counts {} (no arcs), 12, 14, 23, 34, 12-34, 14-23. It does not count 13 because 2 is an isolated point. - David Callan, Sep 18 2007
In my 2003 paper I introduced L-algebras. These are K-vector spaces equipped with two binary operations > and < satisfying (x > y) < z = x > (y < z). In my arXiv paper math-ph/0709.3453 I show that the free L-algebra on one generator is related to symmetric ternary trees with odd degrees, so the dimensions of the homogeneous components are 1, 2, 7, 30, 143, .... These L-algebras are closely related to the so-called triplicial-algebras, 3 associative operations and 3 relations whose free object is related to even trees. - Philippe Leroux (ph_ler_math(AT)yahoo.com), Oct 05 2007
a(n-1) is also the number of projective dependency trees with n nodes. - Marco Kuhlmann (marco.kuhlmann(AT)lingfil.uu.se), Apr 06 2010
Number of subpartitions of [1^2, 2^2, ..., n^2]. - Franklin T. Adams-Watters, Apr 13 2011
a(n) = sum of (n+1)-th row terms of triangle A143603. - Gary W. Adamson, Jul 07 2011
Also the number of Dyck n-paths with up steps colored in two ways (N or A) and avoiding NA. The 7 Dyck 2-paths are NDND, ADND, NDAD, ADAD, NNDD, ANDD, and AADD. - David Scambler, Jun 24 2013
a(n) is also the number of permutations avoiding 213 in the classical sense which can be realized as labels on an increasing strict binary tree with 2n-1 nodes. See A245904 for more information on increasing strict binary trees. - Manda Riehl Aug 07 2014
With offset 1, a(n) is the number of ordered trees (A000108) with n non-leaf vertices and n leaf vertices such that every non-leaf vertex has a leaf child (and hence exactly one leaf child). - David Callan, Aug 20 2014
a(n) is the number of paths in the plane with unit east and north steps, never going above the line x=2y, from (0,0) to (2n+1,n). - Ira M. Gessel, Jan 04 2018
a(n) is the number of words on the alphabet [n+1] that avoid the patterns 231 and 221 and contain exactly one 1 and exactly two occurrences of every other letter. - Colin Defant, Sep 26 2018
a(n) is the number of Motzkin paths of length 3n with n of each type of step, such that (1, 1) and (1, 0) alternate (ignoring (-1, 1) steps). All paths start with a (1, 1) step. - Helmut Prodinger, Apr 08 2019
Hankel transform omitting a(0) is A051255(n+1). - Michael Somos, May 15 2022
If f(x) is the generating function for (-1)^n*a(n), a real solution of the equation y^3 - y^2 - x = 0 is given by y = 1 + x*f(x). In particular 1 + f(1) is Narayana's cow constant, A092526, aka the "supergolden" ratio. - R. James Evans, Aug 09 2023
This is instance k = 2 of the family {c(k, n+1)}A130564.%20_Wolfdieter%20Lang">{n>=0} described in A130564. _Wolfdieter Lang, Feb 04 2024
Also the number of quadrangulations of a (2n+4)-gon which do not contain any diagonals incident to a fixed vertex. - Esther Banaian, Mar 12 2025

			a(3) = 30 since the top row of Q^3 = (12, 12, 5, 1).
G.f. = 1 + 2*x + 7*x^2 + 30*x^3 + 143*x^4 + 728*x^5 + 3876*x^6 + 21318*x^7 + ... - _Michael Somos_, May 15 2022

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

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These are the odd indices of A047749.
Cf. A121645, A115728, A143603, A236194.
Cf. A000260, A007318, A024485, A051255, A071948, A092526, A110616, A258708.
Cf. A305574 (the same with offset 1 and the initial 1 replaced with 5).
Cf. A130564 (comment on c(k, n+1)).

a006013 n = a007318 (3 * n + 1) n `div` (n + 1)
a006013' n = a258708 (2 * n + 1) n
-- Reinhard Zumkeller, Jun 22 2015

[Binomial(3*n+1,n)/(n+1): n in [0..30]]; // Vincenzo Librandi, Mar 29 2015

Binomial[3#+1,#]/(#+1)&/@Range[0,30]  (* Harvey P. Dale, Mar 16 2011 *)

A006013(n) = binomial(3*n+1,n)/(n+1) \\ M. F. Hasler, Jan 08 2024

from math import comb
def A006013(n): return comb(3*n+1,n)//(n+1) # Chai Wah Wu, Jul 30 2022

def A006013_list(n) :
    D = [0]*(n+1); D[1] = 1
    R = []; b = false; h = 1
    for i in range(2*n) :
        for k in (1..h) : D[k] += D[k-1]
        if b : R.append(D[h]); h += 1
        b = not b
    return R
A006013_list(23) # Peter Luschny, May 03 2012

G.f. is square of g.f. for ternary trees, A001764 [Knuth, 2014]. - N. J. A. Sloane, Dec 09 2014
Convolution of A001764 with itself: 2*C(3*n + 2, n)/(3*n + 2), or C(3*n + 2, n+1)/(3*n + 2).
G.f.: (4/(3*x)) * sin((1/3)*arcsin(sqrt(27*x/4)))^2.
G.f.: A(x)/x with A(x)=x/(1-A(x))^2. - Vladimir Kruchinin, Dec 26 2010
From Gary W. Adamson, Jul 14 2011: (Start)
a(n) is the top left term in M^n, where M is the infinite square production matrix:
   2, 1, 0, 0, 0, ...
   3, 2, 1, 0, 0, ...
   4, 3, 2, 1, 0, ...
   5, 4, 3, 2, 1, ...
   ... (End)
From Gary W. Adamson, Aug 11 2011: (Start)
a(n) is the sum of top row terms in Q^n, where Q is the infinite square production matrix as follows:
   1, 1, 0, 0, 0, ...
   2, 2, 1, 0, 0, ...
   3, 3, 2, 1, 0, ...
   4, 4, 3, 2, 1, ...
   ... (End)
D-finite with recurrence: 2*(n+1)*(2n+1)*a(n) = 3*(3n-1)*(3n+1)*a(n-1). - R. J. Mathar, Dec 17 2011
a(n) = 2*A236194(n)/n for n > 0. - Bruno Berselli, Jan 20 2014
a(n) = A258708(2*n+1, n). - Reinhard Zumkeller, Jun 22 2015
From Ilya Gutkovskiy, Dec 29 2016: (Start)
E.g.f.: 2F2([2/3, 4/3]; [3/2,2]; 27*x/4).
a(n) ~ 3^(3*n+3/2)/(sqrt(Pi)*4^(n+1)*n^(3/2)). (End)
a(n) = A110616(n+1, 1). - Ira M. Gessel, Jan 04 2018
0 = v0*(+98415*v2 -122472*v3 +32340*v4) +v1*(+444*v3 -2968*v4) +v2*(-60*v2 +56*v3 +64*v4) where v0=a(n)^2, v1=a(n)*a(n+1), v2=a(n+1)^2, v3=a(n+1)*a(n+2), v4=a(n+2)^2 for all n in Z if a(-1)=-2/3 and a(n)=0 for n<-1. - Michael Somos, May 15 2022
a(n) = (1/4^n) * Product_{1 <= i <= j <= 2*n} (2*i + j + 2)/(2*i + j - 1). Cf. A000260. - Peter Bala, Feb 21 2023
From Karol A. Penson, Jun 02 2023: (Start)
a(n) = Integral_{x=0..27/4} x^n*W(x) dx, where
W(x) = (((9 + sqrt(81 - 12*x))^(2/3) - (9 - sqrt(81 - 12*x))^(2/3)) * 2^(1/3) * 3^(1/6)) / (12 * Pi * x^(1/3)), for x in (0, 27/4).
This integral representation is unique as W(x) is the solution of the Hausdorff power moment problem. Using only the definition of a(n), W(x) can be proven to be positive. W(x) is singular at x = 0, with the singularity x^(-1/3), and for x > 0 is monotonically decreasing to zero at x = 27/4. At x = 27/4 the first derivative of W(x) is infinite. (End)
G.f.: hypergeometric([2/3,1,4/3], [3/2,2], (3^3/2^2)*x). See the e.g.f. above. - Wolfdieter Lang, Feb 04 2024
a(n) = A024485(n+1)/3. - Michael Somos, Oct 14 2024
G.f.: (Sum_{n >= 0} binomial(3*n+2, n)*x^n) / (Sum_{n >= 0} binomial(3*n, n)*x^n) = (B(x) - 1)/(x*B(x)), where B(x) = Sum_{n >= 0} binomial(3*n, n)/(2*n+1) * x^n is the g.f. of A001764. - Peter Bala, Dec 13 2024
The g.f. A(x) is uniquely determined by the conditions A(0) = 1 and [x^n] A(x)^(-n) = -2 for all n >= 1. Cf. A006632. - Peter Bala, Jul 24 2025

Edited by N. J. A. Sloane, Feb 21 2008

A027610 The number of Apollonian networks (planar 3-trees) with n+3 vertices.

Original entry on oeis.org

1, 1, 1, 3, 7, 24, 93, 434, 2110, 11002, 58713, 321776, 1792133, 10131027, 57949430, 334970205, 1953890318, 11489753730, 68054102361, 405715557048, 2433003221232, 14668536954744, 88869466378593, 540834155878536, 3304961537938269, 20273202069859769

Offset: 1

Gordon F. Royle

Previous name was: Number of chordal planar triangulations; also number of planar triangulations with maximal number of triangles; also number of graphs obtained from the tetrahedron by repeatedly inserting vertices of degree 3 into a triangular face; also number of uniquely 4-colorable planar graphs; also number of simplicial 3-clusters with n cells; also Apollonian networks with n+3 vertices.
Also arises in enumeration of spectral isomers of alkane systems (see Cyvin et al.). - N. J. A. Sloane, Aug 15 2006
Chordal planar triangulations: take planar triangulations on n nodes, divide them into classes according to how many triangles they contain (all have 2n-4 triangular faces but may have additional triangles); count triangulations in class with most triangles.
If mirror images are not taken as equivalent, A007173 is obtained instead. - Brendan McKay, Mar 08 2014
Number of unoriented polyominoes composed of n tetrahedral cells of the hyperbolic regular tiling with Schläfli symbol {3,3,oo}. For unoriented polyominoes, chiral pairs are counted as one. - Robert A. Russell, Mar 20 2024

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

Vincenzo Librandi, Table of n, a(n) for n = 1..200
L. W. Beineke and R. E. Pippert Enumerating dissectable polyhedra by their automorphism groups, Can. J. Math., 26 (1974), 50-67.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
CombOS - Combinatorial Object Server, generate planar graphs
S. J. Cyvin, Jianji Wang, J. Brunvoll, Shiming Cao, Ying Li, B. N. Cyvin, and Yugang Wang, Staggered conformers of alkanes: complete solution of the enumeration problem, J. Molec. Struct. 413-414 (1997), 227-239.
Paul Jungeblut, Edge Guarding Plane Graphs, Master Thesis, Karlsruhe Institute of Technology (Germany, 2019).
F. Hering et al., The enumeration of stack polytopes and simplicial clusters, Discrete Math., 40 (1982), 203-217.
Manfred Scheucher, Hendrik Schrezenmaier, Raphael Steiner, A Note On Universal Point Sets for Planar Graphs, arXiv:1811.06482 [math.CO], 2018.

Sum of A047776, A047775, A047774, A047773, A047762, A047760, A047758, A047754, A047753, A047752, A047751, A047771, A047769, A047766 (twice), A047765, A047764.

Cf. A007173 (oriented), A371350 (chiral), A371351 (achiral), A001764 (rooted), A000207 {3,oo}, A182322 {3,3,3,oo}.

A001764 := proc(m) RETURN((3*m)!/(m!*(2*m+1)!)); end; # Gives A001764(m)
A047749 := proc(m) local x; if m mod 2 = 1 then x := (m-1)/2; RETURN((3*x+1)!/((x+1)!*(2*x+1)!)); fi; RETURN(A001764(m/2)); end; # Gives A047749(m)
A027610 := proc(n) local N; N := 0; N := N + A001764(n)/(12*(n+1)); if n mod 2 = 0 then N := N + 5/24*A001764(n/2); fi; if (n-1) mod 3 = 0 then N := N + 1/3*A001764((n-1)/3); fi; if (n-1) mod 4 = 0 then N := N + 1/4*A001764((n-1)/4); fi;
if (n-2) mod 6 = 0 then N := N + 1/6*A001764((n-2)/6); fi; N := N + 3/8*A047749(n); if (2*n-1) mod 3 = 0 then N := N + 1/6*A047749((2*n-1)/3); fi; RETURN(N); end;

Table[Binomial[3 n, 2 n]/(6 (2 n + 1) (2 n + 2)) + If[EvenQ[n], 7 Binomial[3 n/2, n]/(12 (n + 1)), 3 Binomial[3 n/2 - 1/2, n]/(4 (n + 1))] + Switch[Mod[n, 3], 1, Binomial[n - 1, 2 n/3 - 2/3]/(2 n/3 + 1/3), 2, Binomial[n - 1, 2 n/3 - 1/3]/(2 n/3 + 2/3), , 0]/3 + If[1 == Mod[n,4], Binomial[3 n/4 - 3/4, n/2 - 1/2]/(n/2 + 1/2), 0]/4 + If[2 == Mod[n, 6], Binomial[n/2 - 1, n/3 - 2/3]/(n/3 + 1/3), 0]/6, {n, 1, 30}] (* _Robert A. Russell, Apr 11 2012 *)

T(m)={if(m<0||denominator(m)!=1,0,(3*m)!/(m!*(2*m+1)!))};
U(k)={if(k<0||denominator(k)!=1,0,if(k%2,my(m=(k-1)/2);(3*m+1)!/((m+1)!*(2*m+1)!),T(k/2)))};
S(n)=T(n)/(12*(n+1))+5*T(n/2)/24+T((n-1)/3)/3+T((n-1)/4)/4+T((n-2)/6)/6+3*U(n)/8+U((2*n-1)/3)/6;
for(k=1,26,print1(S(k),", ")) \\ Hugo Pfoertner, Mar 07 2020

From Robert A. Russell, Mar 20 2024: (Start)
a(n) = C(3n,n)/(6*(2n+1)*(2n+2)) + ([0==n mod 2]*7*C(3n/2,n) + [1==n mod 2]*9*C((3n-1)/2,n)) / (12(n+1)) + [1==n mod 3]*C(n-1,(n-1)/3)/(2n+1) + [2==n mod 3]*C(n-1,(n-2)/3)/(2n+2) + [1==n mod 4]*C((3n-3)/4,(n-1)/2)/(2n+2) + [2==n mod 6]*C(n/2-1,(n-2)/3)/(2n+2).
a(n) = A007173(n) - A371350(n) = (A007173(n) + A371351(n))/2 = A371350(n) + A371351(n).
a(n) = h(3,n) in Table 8 of Hering link.
G.f.: (-16 + 4*G(z) - 2*G(z)^2 + z*G(z)^4 + 14*G(z^2) + 9z*G(z^2)^2 + 8z*G(z^3) + 4z^2*G(z^3)^2 + 6z*G(z^4) + 4z^2*G(z^6))/24, where G(z) = 1 + z*G(z)^3 is the g.f. for A001764. (End)

One additional term from Robert A. Russell, Apr 11 2012
Noted the name "Apollonian network" by Brendan McKay, Mar 08 2014
New name from Allan Bickle, Feb 21 2024

A143546 G.f. A(x) satisfies A(x) = 1 + xA(x)^3A(-x)^2.

Original entry on oeis.org

1, 1, 1, 3, 5, 18, 35, 136, 285, 1155, 2530, 10530, 23751, 100688, 231880, 996336, 2330445, 10116873, 23950355, 104819165, 250543370, 1103722620, 2658968130, 11777187240, 28558343775, 127067830773, 309831575760, 1383914371728, 3390416787880, 15194457001440

Offset: 0

Paul D. Hanna, Aug 23 2008

Number of achiral polyominoes composed of n hexagonal cells of the hyperbolic regular tiling with Schläfli symbol {6,oo}. A stereographic projection of the {6,oo} tiling on the Poincaré disk can be obtained via the Christensson link. - Robert A. Russell, Jan 23 2024

Number of achiral noncrossing partitions composed of n blocks of size 5. - Andrew Howroyd, Feb 08 2024

			G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 5*x^4 + 18*x^5 + 35*x^6 + 136*x^7 + ...
A(x) = 1 + x*A(x)^3*A(-x)^2 where
A(x)^3 = 1 + 3x + 6x^2 + 16x^3 + 39x^4 + 114x^5 + 304x^6 + 936x^7 + ...
A(-x)^2 = 1 - 2x + 3x^2 - 8x^3 + 17x^4 - 52x^5 + 125x^6 - 408x^7 + ...
Also, A(x) = G(x^2) + x*G(x^2)^3 where
G(x) = 1 + x + 5*x^2 + 35*x^3 + 285*x^4 + 2530*x^5 + 23751*x^6 + ...
G(x)^3 = 1 + 3*x + 18*x^2 + 136*x^3 + 1155*x^4 + 10530*x^5 + ...

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Michel Bousquet and Cédric Lamathe, On symmetric structures of order two, Discrete Math. Theor. Comput. Sci. 10 (2008), 153-176. See Table 1. - From _N. J. A. Sloane_, Jul 12 2011
Malin Christensson, Make hyperbolic tilings of images, web page, 2019.

Column k=5 of A369929 and k=6 of A370062.
Cf. A118970.
Polyominoes: A221184(n-1) (oriented), A004127 (unoriented), A369473 (chiral), A002294 (rooted), A047749 {4,oo}, A369472 {5,oo}.

terms = 28;
A[] = 1; Do[A[x] = 1 + x A[x]^3 A[-x]^2 + O[x]^terms // Normal, {terms}];
CoefficientList[A[x], x] (* Jean-François Alcover, Jul 24 2018 *)
p=6; Table[If[EvenQ[n],Binomial[(p-1)n/2,n/2]/((p-2)n/2+1),If[OddQ[p],(p-1)Binomial[(p-1)n/2-1,(n-1)/2]/((p-2)n+1),p Binomial[(p-1)n/2-1/2,(n-1)/2]/((p-2)n+2)]],{n,0,35}] (* Robert A. Russell, Jan 23 2024 *)

{a(n)=my(A=1+O(x^(n+1)));for(i=0,n,A=1+x*A^3*subst(A^2,x,-x));polcoef(A,n)}

{a(n)=my(m=n\2,p=2*(n%2)+1);binomial(5*m+p-1,m)*p/(4*m+p)}

G.f.: A(x) = G(x^2) + x*G(x^2)^3 where G(x) = 1 + x*G(x)^5 is the g.f. of A002294.
a(2n) = binomial(5*n,n)/(4*n+1); a(2n+1) = binomial(5*n+2,n)*3/(4*n+3).
From Robert A. Russell, Jan 23 2024: (Start)
a(n+2)/a(n) ~ 3125/256. a(2m+1)/a(2m) ~ 75/16; a(2m)/a(2m-1) ~ 125/48.
a(n) = 2*A004127(n) - A221184(n-1) = A221184(n-1) - 2*A369473(n) = A004127(n) - A369473(n). (End)
a(2m) = A002294(m) ~ (5^5/4^4)^m*sqrt(5/(2*Pi*(4*m)^3)). - Robert A. Russell, Jul 15 2024
From Seiichi Manyama, Jul 07 2025: (Start)
G.f. A(x) satisfies A(x)*A(-x) = (A(x) + A(-x))/2 = G(x^2), where G(x) = 1 + x*G(x)^5 is the g.f. of A002294.
a(0) = 1; a(n) = Sum_{i, j, k>=0 and i+2*j+2*k=n-1} a(i) * a(2*j) * a(2*k). (End)
a(0) = 1; a(n) = Sum_{i, j, k, l, m>=0 and i+j+k+l+m=n-1} (-1)^(i+j) * a(i) * a(j) * a(k) * a(l) * a(m). - Seiichi Manyama, Jul 08 2025

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

A047751 Number of dissectable polyhedra with n tetrahedral cells and symmetry of type K.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 30, 0, 0

Offset: 1

N. J. A. Sloane

nonn

One of 17 different symmetry types comprising A007173 and A027610 and one of 10 for A371351. Also the number of tetrahedral clusters or polyominoes of the regular tiling with Schläfli symbol {3,3,oo}, both having type K achiral symmetry and n tetrahedral cells. The center of symmetry is the center of a tetrahedral cell (3); the order of the symmetry group is 24. An achiral polyomino is identical to its reflection. - Robert A. Russell, Mar 22 2024

L. W. Beineke and R. E. Pippert, Enumerating dissectable polyhedra by their automorphism groups, Canad. J. Math., 26 (1974), 50-67.
Robert A. Russell, Mathematica Graphics3D program for A047751 examples

Cf. A007173 (oriented), A027610 (unoriented), A371351 (achiral), A001764 (rooted).

Table[Boole[1==n]+Switch[Mod[n,24],5,12Binomial[(n-5)/8,(n-5)/12],17,24Binomial[(n-9)/8,(n-17)/24],,0]/(n+7),{n,60}] (* _Robert A. Russell, Mar 22 2024 *)

a(1)=1, a(n)=0 unless n == 5 (mod 12); a(12m+5) = A047749(m).

G.f.: z + z^5*G(z^24) + z^17*G(z^24)^2, where G(z) = 1 + z*G(z)^3 is the g.f. for A001764. - Robert A. Russell, Mar 22 2024

A005034 Number of nonequivalent dissections of a polygon into n quadrilaterals by nonintersecting diagonals up to rotation.

Original entry on oeis.org

1, 1, 1, 2, 7, 25, 108, 492, 2431, 12371, 65169, 350792, 1926372, 10744924, 60762760, 347653944, 2009690895, 11723100775, 68937782355, 408323229930, 2434289046255, 14598011263089, 88011196469040, 533216750567280, 3245004785069892, 19829768942544276, 121639211516546668

Offset: 0

N. J. A. Sloane

Also, with a different offset, number of colored quivers in the 2-mutation class of a quiver of Dynkin type A_n. - N. J. A. Sloane, Jan 22 2013
Closed formula is given in my paper linked below. - Nikos Apostolakis, Aug 01 2018
Number of oriented 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 oriented polyominoes, chiral pairs are counted as two. - Robert A. Russell, Jan 20 2024

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 290.
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 = 0..1000
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, 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. Leroux and B. Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Quebec 16 (1992), no 1, 53-80.
P. Leroux and B. Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 53-80, 1992. (Annotated scanned copy)
Alexander Stoimenow, On the number of chord diagrams, Discr. Math. 218 (2000), 209-233. See p. 232.
Torkildsen, Hermund A., Colored quivers of type A and the cell-growth problem, J. Algebra and Applications, 12 (2013), #1250133.

Column k=4 of A295224.

Polyominoes: A005036 (unoriented), A369315 (chiral), A047749 (achiral), A385149 (asymmetric), A001764 (rooted), A001683(n+2) {3,oo}, A005038 {5,oo}.

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

a(n) ~ 3^(3*n + 1/2) / (sqrt(Pi) * n^(5/2) * 2^(2*n + 3)). - Vaclav Kotesovec, Mar 13 2016
a(n) = A005036(n) + A369315(n) = 2*A005036(n) - A047749(n) = 2*A369315(n) + A047749(n). - Robert A. Russell, Jan 19 2024
G.f.: (3*G(z) - G(z)^2 + 2*G(z^2) + z*G(z^2)^2 + 2z*G(z^4)) / 4, where G(z)=1+z*G(z)^3 is the g.f. for A001764. - Robert A. Russell, Jun 19 2025

Name clarified by Andrew Howroyd, Nov 20 2017

A208355 Right edge of the triangle in A208101.

Original entry on oeis.org

1, 1, 1, 2, 2, 5, 5, 14, 14, 42, 42, 132, 132, 429, 429, 1430, 1430, 4862, 4862, 16796, 16796, 58786, 58786, 208012, 208012, 742900, 742900, 2674440, 2674440, 9694845, 9694845, 35357670, 35357670, 129644790, 129644790, 477638700, 477638700, 1767263190

Offset: 0

Reinhard Zumkeller, Mar 04 2012

nonn

Number of achiral polyominoes composed of n+1 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. An achiral polyomino is identical to its reflection. - Robert A. Russell, Jan 20 2024

			a(0)=1; a(1)=1; a(2)=1; a(3)=2. - _Robert A. Russell_, Jan 19 2024
____      ________
\  /  /\  \  /\  /  /\     /\
 \/  /__\  \/__\/  /__\   /__\____
     \  /         /\  /\  \  /\  /
      \/         /__\/__\  \/__\/

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Andrei Asinowski, Cyril Banderier, and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).
Malin Christensson, Make hyperbolic tilings of images, web page, 2019.
D. Levin, L. Pudwell, M. Riehl, and A. Sandberg, Pattern Avoidance on k-ary Heaps, Slides of Talk, 2014.
Zhicong Lin, David G. L. Wang, and Tongyuan Zhao, A decomposition of ballot permutations, pattern avoidance and Gessel walks, arXiv:2103.04599 [math.CO], 2021.

Cf. A099363, A208101.

Polyominoes: A001683(n+2) (oriented), A000207 (unoriented). A369314 (chiral), A000108 (rooted), A047749 ({4,oo}).

a208355 n = a208101 n n
a208355_list = map last a208101_tabl

[Ceiling(Catalan(n div 2)): n in [1..40]]; // Vincenzo Librandi, Feb 18 2014

A208355_list := proc(len) local D, b, h, R, i, k;
    D := [seq(0, j=0..len+2)]; D[1] := 1; b := true; h := 2; R := NULL;
    for i from 1 to 2*len do
        if b then
            for k from h by -1 to 2 do D[k] := D[k] - D[k-1] od;
            h := h + 1; R := R, abs(D[2]);
        else
            for k from 1 by 1 to h do D[k] := D[k] + D[k+1] od;
        fi;
        b := not b:
    od;
    return R
end:
A208355_list(38); # Peter Luschny, Dec 19 2017

T[, 0] = 1; T[n, 1] := n; T[n_, n_] := T[n - 1, n - 2]; T[n_, k_] /; 1 < k < n := T[n, k] = T[n - 1, k] + T[n - 1, k - 2];
a[n_] := T[n, n];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 03 2018, from A208101 *)
Table[If[EvenQ[n], Binomial[n,n/2]/(n/2+1), Binomial[n+1,(n+1)/2]/((n+3)/2)], {n,0,40}] (* Robert A. Russell, Jan 19 2024 *)

a(n) = A000108(floor((n+1)/2)), where A000108 = Catalan numbers.
a(n) = A208101(n,n).
a(n) = abs(A099363(n)).
Conjecture: -(n+3)*(n-2)*a(n) - 4*a(n-1) + 4*(n-1)^2*a(n-2) = 0. - R. J. Mathar, Aug 04 2015
From Robert A. Russell, Jan 19 2024: (Start)
a(2m) = C(2m,m)/(m+1); a(2m-1) = a(2m); a(n+2)/a(n) ~ 4.
a(n-1) = 2*A000207(n) - A001683(n+2) = A001683(n+2) - 2*A369314(n) = A000207(n) - A369314(n). (End)
G.f.: (G(z^2)+z*G(z^2)-1)/z, where G(z)=1+z*G(z)^2, the generating function for A000108. - Robert A. Russell, Jan 26 2024
G.f.: ((((1+z)*(1-sqrt(1-4*z^2)))/(2*z^2))-1)/z. - Robert A. Russell, Jan 28 2024
From Peter Bala, Feb 05 2024: (Start)
G.f.: 1/(1 + 2*x) * c(x/(1 + 2*x))^3, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108.
a(n) = Sum_{k = 0..n} (-2)^(n-k)*binomial(n, k)*A000245(k+1).
a(n) = (-2)^n * hypergeom([-n, 3/2, 2], [1, 4], 2). (End)

A047753 Number of dissectable polyhedra with n tetrahedral cells and symmetry of type I.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 7, 0, 0, 0, 12, 0, 0, 0, 29, 0, 0, 0, 55, 0, 0, 0, 143, 0, 0, 0, 271, 0, 0, 0, 728, 0, 0, 0, 1428, 0, 0, 0, 3873, 0, 0, 0, 7752, 0, 0, 0, 21318, 0, 0, 0, 43256, 0, 0, 0, 120175, 0, 0, 0, 246675, 0, 0, 0

Offset: 1

N. J. A. Sloane

nonn

One of 17 different symmetry types comprising A007173 and A027610 and one of 10 for A371351. Also the number of tetrahedral clusters or polyominoes of the regular tiling with Schläfli symbol {3,3,oo}, both having type I achiral symmetry and n tetrahedral cells. The center of symmetry is the center of a tetrahedral cell (3.1); the order of the symmetry group is 8. An achiral polyomino is identical to its reflection. - Robert A. Russell, Mar 22 2024

L. W. Beineke and R. E. Pippert, Enumerating dissectable polyhedra by their automorphism groups, Canad. J. Math., 26 (1974), 50-67.
Robert A. Russell, Mathematica Graphics3D program for A047753 example.

Cf. A047767.

Cf. A007173 (oriented), A027610 (unoriented), A371351 (achiral), A001764 (rooted), A047751 (type K), A047749 (type U).

Table[Switch[Mod[n,8],1,4Binomial[(3n-3)/8,(n-1)/8]/(n+3)-If[17==Mod[n,24],24Binomial[(n-9)/8,(n-17)/24]/(n+7),0],5,4Binomial[(3n-7)/8,(n+3)/8]/(n-1)-If[5==Mod[n,24],12Binomial[(n-5)/8,(n-5)/12]/(n+7),0],,0]-Boole[1==n],{n,50}] (* _Robert A. Russell, Mar 22 2024 *)

If n=4m+1 then A047749(m) - A047751(n), otherwise 0.

G.f.: z*G(z^8) + z^5*G(z^8)^2 - z - z^5*G(z^24) - z^17*G(z^24)^2, where G(z) = 1 + z*G(z)^3 is the g.f. for A001764. - Robert A. Russell, Mar 22 2024

A047764 Number of dissectable polyhedra with n tetrahedral cells and symmetry of type Q.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 30, 0, 0, 0, 0, 0, 55, 0, 0, 0, 0, 0, 143, 0, 0, 0, 0, 0, 273, 0, 0, 0, 0, 0, 728, 0, 0, 0, 0, 0, 1428, 0, 0, 0, 0, 0, 3876, 0, 0, 0, 0, 0, 7752

Offset: 1

N. J. A. Sloane

nonn

One of 17 different symmetry types comprising A007173 and A027610 and one of 10 for A371351. Also the number of tetrahedral clusters or polyominoes of the regular tiling with Schläfli symbol {3,3,oo}, both having type Q achiral symmetry and n tetrahedral cells. The center of symmetry is the center of a tetrahedral face (2.10); the order of the symmetry group is 12. An achiral polyomino is identical to its reflection. - Robert A. Russell, Mar 22 2024

L. W. Beineke and R. E. Pippert, Enumerating dissectable polyhedra by their automorphism groups, Canad. J. Math., 26 (1974), 50-67.
Robert A. Russell, Mathematica Graphics3D program for A047764 examples

Cf. A007173 (oriented), A027610 (unoriented), A371351 (achiral), A001764 (rooted), A047749 (type U).

Table[Switch[Mod[n,12],2,6Binomial[(n-2)/4,(n-2)/12],8,12Binomial[(n-4)/4,(n-2)/6],,0]/(n+4),{n,50}] (* _Robert A. Russell, Mar 22 2024 *)

If n=6m+2 then A047749((n-2)/6), otherwise 0.

G.f.: z^2*G(z^12) + z^8*G(z^12)^2, where G(z) = 1 + z*G(z)^3 is the g.f. for A001764. - Robert A. Russell, Mar 22 2024

cp's OEIS Frontend

A001764 a(n) = binomial(3*n,n)/(2*n+1) (enumerates ternary trees and also noncrossing trees).

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A006013 a(n) = binomial(3*n+1,n)/(n+1).

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A027610 The number of Apollonian networks (planar 3-trees) with n+3 vertices.

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A143546 G.f. A(x) satisfies A(x) = 1 + x*A(x)^3*A(-x)^2.

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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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A001764 a(n) = binomial(3n,n)/(2n+1) (enumerates ternary trees and also noncrossing trees).

A143546 G.f. A(x) satisfies A(x) = 1 + xA(x)^3A(-x)^2.