A002293 -id:A002293 - 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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Jian Zhou, Fat and Thin Emergent Geometries of Hermitian One-Matrix Models, arXiv:1810.03883 [math-ph], 2018.
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
a(n) = 1/(n+1) * [x^n] C(x)^(n+1), where C(x) = (1 - sqrt( 1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. - Peter Bala, Sep 08 2025

A002294 a(n) = binomial(5n, n)/(4n + 1).

Original entry on oeis.org

1, 1, 5, 35, 285, 2530, 23751, 231880, 2330445, 23950355, 250543370, 2658968130, 28558343775, 309831575760, 3390416787880, 37377257159280, 414741863546285, 4628362722856425, 51912988256282175, 584909606696793885, 6617078646960613370

Offset: 0

N. J. A. Sloane

From Wolfdieter Lang, Sep 14 2007: (Start)
a(n), n >= 1, enumerates quintic trees (rooted, ordered, incomplete) with n vertices (including the root).
This is the Pfaff-Fuss-Catalan sequence C^{m}_n for m = 5. See the Graham et al. reference, p. 347. eq. 7.66. See also the Pólya-Szegő reference.
Also 5-Raney sequence. See the Graham et al. reference, pp. 346-347. (End)
a(n) = A258708(3*n, 2*n) for n > 0. - Reinhard Zumkeller, Jun 23 2015
Conjecturally, a(n) is the number of 4-uniform words on the alphabet [n] that avoid the patterns 231 and 221 (see the Defant and Kravitz link). - Colin Defant, Sep 26 2018
From Stillwell (1995), p. 62: "Eisenstein's Theorem. If y^5 + y = x, then y has a power series expansion y = x - x^5 + 10*x^9/2^1 - 15 * 14 * x^13/3! + 20 * 19 * 18*x^17/4! - ...." - Michael Somos, Sep 19 2019
a(n) is the total number of down steps before the first up step in all 4_1-Dyck paths of length 5*n. A 4_1-Dyck path is a lattice path with steps (1, 4), (1, -1) that starts and ends at y = 0 and stays above the line y = -1. - Sarah Selkirk, May 10 2020
Dropping the first 1 (starting from 1, 5, 35, ... with offset 1), the series reversion gives 1, -5, 15, -35, 70, ... (again offset 1), essentially A000332 and row 5 of A027555. - R. J. Mathar, Aug 17 2023
Number of rooted polyominoes composed of n hexagonal cells of the hyperbolic regular tiling with Schläfli symbol {6,oo}. A rooted polyomino has one external edge identified, and chiral pairs are counted as two. A stereographic projection of the {6,oo} tiling on the Poincaré disk can be obtained via the Christensson link. - Robert A. Russell, Jan 27 2024
This is instance k = 5 of the generalized Catalan family {C(k, n)}_{n>=0} given in a comment of A130564. - Wolfdieter Lang, Feb 05 2024

			There are a(2) = 5 quintic trees (vertex degree <= 5 and 5 possible branchings) with 2 vertices (one of them the root). Adding one more branch (one more vertex) to these five trees yields 5*5 + binomial(5,2) = 35 = a(3) such trees.
G.f. = 1 + x + 5*x^2 + 35*x^3 + 285*x^4 + 2530*x^5 + 23751*x^6 + 231880*x^7 + ...
G.f. = t + t^5 + 5*t^9 + 35*t^13 + 285*t^17 + 2530*t^21 + 23751*t^25 + 231880*t^29 + ...

Archiv der Mathematik u. Physik, Editor's note: "Über die Bestimmung der Anzahl der verschiedenen Arten, auf welche sich ein n-Eck durch Diagonalen in lauter m-Ecke zerlegen laesst, mit Bezug auf einige Abhandlungen der Herren Lame, Rodrigues, Binet, Catalan und Duhamel in dem Journal de Mathematiques pures et appliquees, publie par Joseph Liouville. T. III. IV.", Archiv der Mathematik u. Physik, 1 (1841), pp. 193ff; see especially p. 198.
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 23.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, pp. 200, 347.
G. Pólya and G. Szegő, Problems and Theorems in Analysis, Springer-Verlag, Heidelberg, New York, 2 vols., 1972, Vol. 1, problem 211, p. 146 with solution on p. 348.
Ulrike Sattler, Decidable classes of formal power series with nice closure properties, Diplomarbeit im Fach Informatik, Univ. Erlangen - Nürnberg, Jul 27 1994.
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 = 0..100
V. E. Adler and A. B. Shabat, Volterra chain and Catalan numbers, arXiv:1810.13198 [nlin.SI], 2018.
Joerg Arndt, Matters Computational (The Fxtbook), pp. 337-338.
Joerg Arndt, Subset-lex: did we miss an order?, arXiv:1405.6503 [math.CO], 2014-2015.
A. Asinowski, B. Hackl, and S. Selkirk, Down step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.
Paul Barry, Characterizations of the Borel triangle and Borel polynomials, arXiv:2001.08799 [math.CO], 2020.
Paul Barry, On the Gap-sum and Gap-product Sequences of Integer Sequences, arXiv:2104.05593 [math.CO], 2021.
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]
Frits Beukers, Hypergeometric functions, how special are they?, Notices Amer. Math. Soc. 61(1) (2014), 48-56. MR3137256
Wun-Seng Chou, Tian-Xiao He, and Peter J.-S. Shiue, On the Primality of the Generalized Fuss-Catalan Numbers, J. Int. Seqs., Vol. 21 (2018), #18.2.1.
Malin Christensson, Make hyperbolic tilings of images, web page, 2019.
Olivier Danvy, Summa Summarum: Moessner's Theorem without Dynamic Programming, arXiv:2412.03127 [cs.DM], 2024. See p. 25.
Colin Defant and N. Kravitz, Stack-sorting for words, arXiv:1809.09158 [math.CO], 2018.
R. W. Gosper, Rope around the earth
F. Harary, E. M. Palmer and R. C. Read, On the cell-growth problem for arbitrary polygons, Discr. Math. 11 (1975), 371-389.
F. Harary, E. M. Palmer, and R. C. Read, On the cell-growth problem for arbitrary polygons, computer printout, circa 1974.
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.
Forrest M. Hilton, Finite Dynamical Laminations, arXiv:2408.01353 [math.DS], 2024. See p. 7.
V. E. Hoggatt, Jr., 7-page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 287.
R. P. Loh, A. G. Shannon, and A. F. Horadam, Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients, preprint, 1980.
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv:1403.5962 [math.CO], 2014.
Mitchell Paukner, Lucy Pepin, Manda Riehl, and Jarred Wieser, Pattern Avoidance in Task-Precedence Posets, arXiv:1511.00080 [math.CO], 2015-2016.
Karol A. Penson and Karol Zyczkowski, Product of Ginibre matrices : Fuss-Catalan and Raney distribution, arXiv version, arXiv:1103.3453 [math-ph], 2011.
Yuan (Friedrich) Qiu, Joe Sawada, and Aaron Williams, Maximize the Rightmost Digit: Gray Codes for Restricted Growth Strings, WALCOM 2025. See p. 5.
John Stillwell, Eisenstein's footnote, Math. Intelligencer 17(2) (1995), 58-62. MR1336074 (96d:01024)
B. Sury, Generalized Catalan numbers: linear recursion and divisibility, JIS 12 (2009), Article 09.7.5.
L. Takacs, Enumeration of rooted trees and forests, Math. Scientist 18 (1993), 1-10, esp. Eq. (5).
Wikipedia, Fuss-Catalan number.
N. J. Wildberger and Dean Rubine, A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode, Amer. Math. Monthly (2025). See section 12.
S. Yakoubov, Pattern Avoidance in Extensions of Comb-Like Posets, arXiv:1310.2979 [math.CO], 2013-2014.
Jun Yan, Lattice paths enumerations weighted by ascent lengths, arXiv:2501.01152 [math.CO], 2025. See p. 7.

Cf. A001764, A002296, A118969, A258708, A346647 (binomial transform), A346665 (inverse binomial transform).
Fourth column of triangle A062993.
Polyominoes: A221184{n-1} (oriented), A004127 (unoriented), A369473 (chiral), A143546 (achiral), A002293 {5,oo}, A002295 {7,oo}.
Cf. A130564.

List([0..22],n->Binomial(5*n,n)/(4*n+1)); # Muniru A Asiru, Nov 01 2018

a002294 n = a002294_list !! n
a002294_list = [a258708 (3 * n) (2 * n) | n <- [1..]]
-- Reinhard Zumkeller, Jun 23 2015

[ Binomial(5*n,n)/(4*n+1): n in [0..100]]; // Vincenzo Librandi, Mar 24 2011

seq(binomial(5*k+1,k)/(5*k+1),k=0..30); # Robert FERREOL, Apr 03 2015
n:=30:G:=series(RootOf(g = 1+x*g^5, g),x=0,n+1):seq(coeff(G,x,k),k=0..n); # Robert FERREOL, Apr 03 2015

CoefficientList[InverseSeries[ Series[ y - y^5, {y, 0, 100}], x], x][[Range[2, 100, 4]]]
Table[Binomial[5n,n]/(4n+1),{n,0,20}] (* Harvey P. Dale, Dec 30 2011 *)
a[ n_] := SeriesCoefficient[ HypergeometricPFQ[ {1, 2, 3, 4}/5, {2, 3, 5}/4, x 5^5/4^4], {x, 0, n}]; (* Michael Somos, May 06 2015 *)
a[ n_] := With[{m = 4 n + 1}, SeriesCoefficient[ InverseSeries @ Series[ x - x^5, {x, 0, m}], {x, 0, m}]]; (* Michael Somos, May 06 2015 *)

{a(n) = binomial( 5 * n, n) / (4*n + 1)}; /* Michael Somos, Mar 17 2011 */

{a(n) = if( n<0, 0, n = 4*n + 1; polcoeff( serreverse( x - x^5 + x * O(x^n) ), n))}; /* Michael Somos, Mar 17 2011 */

For the connection with the solution of the quintic, hypergeometric series, and Lagrange inversion, see Beukers (2014). - N. J. A. Sloane, Mar 12 2014
G.f.: hypergeometric([1, 2, 3, 4] / 5, [2, 3, 5] / 4, x * 5^5 / 4^4). - Michael Somos, Mar 17 2011
O.g.f. A(x) satisfies A(x) = 1 + x * A(x)^5 = 1 / (1 - x * A(x)^4).
Given g.f. A(x) then z = t * A(t^4) satisfies 0 = z^5 - z + t. - Michael Somos, Mar 17 2011
a(n) = binomial(5*n, n - 1)/n, n >= 1, a(0) = 1. From the Lagrange series of the o.g.f. A(x) with its above given implicit equation.
a(n) = upper left term in M^n, M = the production matrix:
   1,  1;
   4,  4,  1;
  10, 10,  4,  1;
  20, 20, 10,  4,  1;
  ...
where (1, 4, 10, 20, ...) is the tetrahedral sequence, A000292. - Gary W. Adamson, Jul 08 2011
D-finite with recurrence: 8*n*(4*n+1)*(2*n-1)*(4*n-1)*a(n) - 5*(5*n-4)*(5*n-3)*(5*n-2)*(5*n-1)*a(n-1) = 0. - R. J. Mathar, Dec 02 2014
a(n) = binomial(5*n + 1, n)/(5*n + 1) = A062993(n+3,3). - Robert FERREOL, Apr 03 2015
a(0) = 1; a(n) = Sum_{i1 + i2 + ... + i5 = n - 1} a(i1) * a(i2) * ... *a(i5) for n >= 1. - Robert FERREOL, Apr 03 2015
From Ilya Gutkovskiy, Jan 15 2017: (Start)
O.g.f.: 5F4([1/5, 2/5, 3/5, 4/5, 1]; [1/2, 3/4, 1, 5/4]; 3125*x/256).[Cancellation of the 1s, see G.f. the above. - Wolfdieter Lang, Feb 05 2024]
E.g.f.: 4F4([1/5, 2/5, 3/5, 4/5]; [1/2, 3/4, 1, 5/4]; 3125*x/256).
a(n) ~ 5^(5*n + 1/2)/(sqrt(Pi) * 2^(8*n + 7/2) * n^(3/2)). (End)
x*A'(x)/A(x) = (A(x) - 1)/(- 4*A(x) + 5) = x + 9*x^2 + 91*x^3 + 969*x^4 + ... is the o.g.f. of A163456. Cf. A001764 and A002293 - A002296. - Peter Bala, Feb 04 2022
G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^9). - Seiichi Manyama, Jun 16 2025
a(n) = (1/(3*n+1)) * [x^n] C(x)^(3*n+1), where C(x) = (1 - sqrt( 1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. - Peter Bala, Sep 08 2025

More terms from Olivier Gérard, Jul 05 2001

A002295 Number of dissections of a polygon: binomial(6n,n)/(5n+1).

Original entry on oeis.org

1, 1, 6, 51, 506, 5481, 62832, 749398, 9203634, 115607310, 1478314266, 19180049928, 251857119696, 3340843549855, 44700485049720, 602574657427116, 8175951659117794, 111572030260242090, 1530312970340384580, 21085148778264281865, 291705220704719165526

Offset: 0

N. J. A. Sloane

From Wolfdieter Lang, Sep 14 2007: (Start)
a(n), n >= 1, enumerates sextic (6-ary) trees (rooted, ordered, incomplete) with n vertices (including the root).
Pfaff-Fuss-Catalan sequence C^{m}_n for m=6. See the Graham et al. reference, p. 347. eq. 7.66. See also the Pólya-Szegő reference.
Also 6-Raney sequence. See the Graham et al. reference, p. 346-7. (End)
This is instance k = 6 of the generalized Catalan family {C(k, n)}A130564.%20-%20_Wolfdieter%20Lang">{n>=0} given in a comment of A130564. - _Wolfdieter Lang, Feb 05 2024

			There are a(2)=6 sextic trees (vertex degree <= 6 and 6 possible branchings) with 2 vertices (one of them the root). Adding one more branch (one more vertex) to these 6 trees yields 6*6 + binomial(6,2) = 51 = a(3) such trees.

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 23.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, pp. 200, 347.
G. Pólya and G. Szegő, Problems and Theorems in Analysis, Springer-Verlag, Heidelberg, New York, 2 vols., 1972, Vol. 1, problem 211, p. 146 with solution on p. 348.
Ulrike Sattler, Decidable classes of formal power series with nice closure properties, Diplomarbeit im Fach Informatik, Univ. Erlangen - Nürnberg, Jul 27 1994
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).
Editor's note: "Über die Bestimmung der Anzahl der verschiedenen Arten, auf welche sich ein n-Eck durch Diagonalen in lauter m-Ecke zerlegen laesst, mit Bezug auf einige Abhandlungen der Herren Lamé, Rodrigues, Binet, Catalan und Duhamel in dem Journal de Mathématiques pures et appliquées, publié par Joseph Liouville. T. III. IV.", Archiv der Mathematik u. Physik, 1 (1841), pp. 193ff; see especially p. 198.

T. D. Noe, Table of n, a(n) for n = 0..100
V. E. Adler and A. B. Shabat, Volterra chain and Catalan numbers, arXiv:1810.13198 [nlin.SI], 2018.
Paul Barry, On the Gap-sum and Gap-product Sequences of Integer Sequences, arXiv:2104.05593 [math.CO], 2021.
Wun-Seng Chou, Tian-Xiao He, and Peter J.-S. Shiue, On the Primality of the Generalized Fuss-Catalan Numbers, J. Int. Seqs., Vol. 21 (2018), #18.2.1.
Olivier Danvy, Summa Summarum: Moessner's Theorem without Dynamic Programming, arXiv:2412.03127 [cs.DM], 2024. See p. 25.
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.
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.
V. E. Hoggatt, Jr., 7-page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 288
R. P. Loh, A. G. Shannon, and A. F. Horadam, Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients, Preprint, 1980.
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014.
Yuan (Friedrich) Qiu, Joe Sawada, and Aaron Williams, Maximize the Rightmost Digit: Gray Codes for Restricted Growth Strings, WALCOM 2025. See p. 5.
B. Sury, Generalized Catalan numbers: linear recursion and divisibility, JIS 12 (2009), Article 09.7.5.
L. Takacs, Enumeration of rooted trees and forests, Math. Scientist 18 (1993), 1-10, esp. Eq. (5).
Jun Yan, Lattice paths enumerations weighted by ascent lengths, arXiv:2501.01152 [math.CO], 2025. See p. 7.

Cf. A001764, A002293, A002294, A002296, A130564.

Fifth column of triangle A062993.

List([0..22],n->Binomial(6*n,n)/(5*n+1)); # Muniru A Asiru, Nov 01 2018

[Binomial(6*n, n)/(5*n + 1): n in [0..20]]; // Vincenzo Librandi, Mar 17 2015

A002295:=n->binomial(6*n, n)/(5*n + 1); seq(A002295(n), n=0..20); # Wesley Ivan Hurt, Jan 29 2014
n:=20:G:=series(RootOf(g = 1+x*g^6, g),x=0,n+1):seq(coeff(G,x,k),k=0..n); # Robert FERREOL, Apr 03 2015

Table[Binomial[6n, n]/(5n + 1), {n, 0, 20}] (* Stefan Steinerberger, Apr 06 2006 *)

A002295(n)=binomial(6*n,n)/(5*n+1) \\ M. F. Hasler, Apr 08 2015

O.g.f.: A(x) = 1 + x*A(x)^6 = 1/(1-x*A(x)^5).
a(n) = binomial(6*n,n-1)/n, n >= 1, a(0)=1. From the Lagrange series of the o.g.f. A(x) with its above given implicit equation.
a(n) = upper left term in M^n, M = the production matrix:
   1,  1
   5,  5,  1
  15, 15,  5,  1
  35, 35, 15,  5,  1
  ...
where (1, 5, 15, 35, ...) = A000332 starting with 1. - Gary W. Adamson, Jul 08 2011
a(n) are special values of Jacobi polynomials, in Maple notation:
  a(n) = JacobiP(n-1, 5*n+1, -n, 1)/n, n=1, 2, ... . - Karol A. Penson, Mar 17 2015
a(n) = binomial(6*n+1, n)/(6*n+1) = A062993(n+4,4). - Robert FERREOL, Apr 03 2015
a(0) = 1; a(n) = Sum_{i1+i2+...+i6=n-1} a(i1)*a(i2)*...*a(i6) for n>=1. - Robert FERREOL, Apr 03 2015
D-finite with recurrence: 5*n*(5*n+1)*(5*n-3)*(5*n-2)*(5*n-1)*a(n) - 72*(6*n-5)*(6*n-1)*(3*n-1)*(2*n-1)*(3*n-2)*a(n-1) = 0. - R. J. Mathar, Sep 06 2016
From Ilya Gutkovskiy, Jan 15 2017: (Start)
O.g.f.: 5F4(1/6,1/3,1/2,2/3,5/6; 2/5,3/5,4/5,6/5; 46656*x/3125).
E.g.f.: 5F5(1/6,1/3,1/2,2/3,5/6; 2/5,3/5,4/5,1,6/5; 46656*x/3125).
a(n) ~ 3^(6*n+1/2)*64^n/(sqrt(Pi)*5^(5*n+3/2)*n^(3/2)). (End)
x*A'(x)/A(x) = (A(x) - 1)/(- 5*A(x) + 6) = x + 11*x^2 + 136*x^3 + 1771*x^4 + ... = (1/6)*Sum_{n >= 1} binomial(6*n,n)*x^n. Cf. A001764 and A002293 - A002296. - Peter Bala, Feb 04 2022
G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^11). - Seiichi Manyama, Jun 16 2025

More terms from Stefan Steinerberger, Apr 06 2006

Edited by M. F. Hasler, Apr 08 2015

A005810 a(n) = binomial(4n,n).

Original entry on oeis.org

1, 4, 28, 220, 1820, 15504, 134596, 1184040, 10518300, 94143280, 847660528, 7669339132, 69668534468, 635013559600, 5804731963800, 53194089192720, 488526937079580, 4495151581425648, 41432089765583440, 382460951663844400

Offset: 0

N. J. A. Sloane

Start off with 0 balls in a box. Find the number of ways you can throw 3 balls back out. Then continue to throw 4 balls into the box after each stage. (I.e., the first stage is 0. Then at the next stage there are 4 ways to throw 3 balls back out.) - Ruppi Rana (ruppirana007(AT)hotmail.com), Mar 03 2004
Central coefficients of A094527. - Paul Barry, Mar 08 2011
This is the case m = 2n in Catalan's formula (2m)!*(2n)!/(m!*(m+n)!*n!) - see Umberto Scarpis in References. - Bruno Berselli, Apr 27 2012
A generating function in terms of a (labyrinthine) solution to a depressed quartic equation is given in the Copeland link for signed A005810. - Tom Copeland, Oct 10 2012
Conjecture: a(n) == 4 (mod n^3) iff n is prime. - Gary Detlefs, Apr 03 2013
For prime p, the congruence a(p) = binomial(4*p,p) = 4 (mod p^3)  is a known generalization of Wolstenholme's theorem. See Mestrovic, Section 6, equation 35. - Peter Bala, Dec 28 2014

			G.f. = 1 + 4*x + 28*x^2 + 220*x^3 + 1820*x^4 + 15504*x^5 + 134596*x^6 + ...

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.
Umberto Scarpis, Sui numeri primi e sui problemi dell'analisi indeterminata in Questioni riguardanti le matematiche elementari, Nicola Zanichelli Editore (1924-1927, third Edition), page 11.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000 (terms 0..100 from T. D. Noe, terms 101..213 from Muniru A Asiru)
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].
Tom Copeland, Discriminating Deltas, Depressed Equations, and Generalized Catalan Numbers, 2012, pp. 5-6.
Romeo Meštrović, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.
Ruppi Rana, Title? [Broken link]

Row 4 of A060539.
binomial(k*n,n): A000984 (k = 2), A005809 (k = 3), A001449 (k = 5), A004355 (k = 6), A004368 (k = 7), A004381 (k = 8), A169958 - A169961 (k = 9 thru 12).
Cf. A007318, A182400, A262261, A378806, A378807.

List([0..20],n->Binomial(4*n,n)); # Muniru A Asiru, Mar 19 2018

a005810 n = a007318 (4*n) n  -- Reinhard Zumkeller, Mar 04 2012

[ Binomial(4*n,n): n in [0..100] ]; // Vincenzo Librandi, Apr 13 2011

seq(binomial(4*n,n),n=0..20); # Muniru A Asiru, Mar 19 2018

Table[Binomial[4n,n],{n,0,19}] (* Geoffrey Critzer, Sep 15 2013 *)

a(n) = binomial(4*n, n); \\ Altug Alkan, Mar 19 2018

from math import comb
def A005810(n): return comb(n<<2,n) # Chai Wah Wu, Aug 01 2023

a(n) is asymptotic to c*(256/27)^n/sqrt(n) with c = sqrt(2 / (3 Pi)) = 0.460658865961780639... - Benoit Cloitre, Jan 26 2003; corrected by Charles R Greathouse IV, Dec 14 2006
a(n) = Sum_{k=0..2n} binomial(2n,k) * binomial(2n,k-n). - Paul Barry, Mar 08 2011
G.f.: g/(4-3*g) where g = 1+x*g^4 is the g.f. of A002293. - Mark van Hoeij, Nov 11 2011
D-finite with recurrence: 3*n*(3*n-1)*(3*n-2)*a(n) - 8*(4*n-3)*(2*n-1)*(4*n-1)*a(n-1) = 0. - R. J. Mathar, Dec 02 2012
a(n) = binomial(4*n,n-1)*(3*n+1)/n. - Gary Detlefs, Apr 03 2013
a(n) = C(4*n-1,n-1)*C(16*n^2,2)/(3*n*C(4*n+1,3)), n>0. - Gary Detlefs, Jan 02 2014
a(n) = Sum_{i,j,k = 0..n} binomial(n,i)*binomial(n,j)*binomial(n,k)* binomial(n,i+j+k). - Peter Bala, Dec 28 2014
a(n) = GegenbauerC(n, -2*n, -1). - Peter Luschny, May 07 2016
From Ilya Gutkovskiy, Nov 22 2016: (Start)
O.g.f.: 3F2(1/4,1/2,3/4; 1/3,2/3; 256*x/27).
E.g.f.: 3F3(1/4,1/2,3/4; 1/3,2/3,1; 256*x/27). (End)
a(n) = hypergeom([-3*n, -1*n], [1], 1). - Peter Luschny, Mar 19 2018
RHS of the identity Sum_{k = 0..2*n} (-1)^(n+k)*binomial(4*n, k)* binomial(4*n, 2*n-k) = binomial(4*n,n). - Peter Bala, Oct 07 2021
From Peter Bala, Feb 20 2022: (Start)
The o.g.f. A(x) satisfies the differential equation
(-256*x^3 + 27*x^2)*A(x)''' + (-1152*x^2 + 54*x)*A(x)'' + (-816*x + 6)*A(x)' - 24*A(x) = 0 with A(0) = 1, A'(0) = 4 and A''(0) = 56.
Algebraic equation: (1 - A(x))*(1 + 3*A(x))^3 +  256*x*A(x)^4 = 0.
Sum_{n >= 1} a(n)*( x*(3*x + 4)^3/(256*(1 + x)^4) )^n = x. (End)
From Amiram Eldar, Dec 07 2024: (Start)
Sum_{n>=1} 1/a(n) = A378806.
Sum_{n>=1} (-1)^n/a(n) = A378807. (End)
From Peter Bala, Jun 29 2025: (Start)
a(n) = (1/8)^n * Sum_{k = n..4*n} binomial(k, n) * binomial(4*n, k).
Sum_{n >= 0 } a(n)*(1/128)^n = (1/5)*(sqrt(2) + sqrt(7 + 5*sqrt(2))). (End)
From Seiichi Manyama, Aug 16 2025: (Start)
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(4*n+1,k).
G.f.: 1/(1 - 4*x*g^3) where g = 1+x*g^4 is the g.f. of A002293. (End)

More terms from Henry Bottomley, Oct 06 2000

Corrected by T. D. Noe, Jan 16 2007

A002296 Number of dissections of a polygon: binomial(7n,n)/(6n+1).

Original entry on oeis.org

1, 1, 7, 70, 819, 10472, 141778, 1997688, 28989675, 430321633, 6503352856, 99726673130, 1547847846090, 24269405074740, 383846168712104, 6116574500860880, 98106248306858715, 1582638261961640247, 25661404527790252375, 417980115131315136400

Offset: 0

N. J. A. Sloane

a(n), n>=1, enumerates heptic (7-ary) trees (rooted, ordered, incomplete) with n vertices (including the root).
Pfaff-Fuss-Catalan sequence C^{m}_n for m=7. See the Graham et al. reference, p. 347. eq. 7.66. See also the Pólya-Szegő reference.
Also 7-Raney sequence. See the Graham et al. reference, pp. 346-347.
a(n) = A258708(3*n,2*n) for n > 0. - Reinhard Zumkeller, Jun 23 2015
This is instance k = 7 of the generalized Catalan family {C(k, n)}A130564.%20-%20_Wolfdieter%20Lang">{n>=0} given in a comment of A130564. - _Wolfdieter Lang, Feb 05 2024

			There are a(2)=7 heptic trees (vertex degree <= 7 and 7 possible branchings) with 2 vertices (one of them the root). Adding one more branch (one more vertex) to these 7 trees yields 7*7 + binomial(7,2) = 70 = a(3) such trees.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, pp. 200, 347.
G. Pólya and G. Szegő, Problems and Theorems in Analysis, Springer-Verlag, Heidelberg, New York, 2 vols., 1972, Vol. 1, problem 211, p. 146 with solution on p. 348.
Ulrike Sattler, Decidable classes of formal power series with nice closure properties, Diplomarbeit im Fach Informatik, Univ. Erlangen - Nürnberg, Jul 27 1994.
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 = 0..100
Wun-Seng Chou, Tian-Xiao He, and Peter J.-S. Shiue, On the Primality of the Generalized Fuss-Catalan Numbers, J. Int. Seqs., Vol. 21 (2018), #18.2.1.
Olivier Danvy, Summa Summarum: Moessner's Theorem without Dynamic Programming, arXiv:2412.03127 [cs.DM], 2024. See p. 25.
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. See Table 6. n = 8. Sequence U(8) p. 387.
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.
V. E. Hoggatt, Jr., 7-page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 289.
R. P. Loh, A. G. Shannon, and A. F. Horadam, Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients, Preprint, 1980.
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv:1403.5962 [math.CO], 2014.
Editor's note: Über die Bestimmung der Anzahl der verschiedenen Arten, auf welche sich ein n-Eck durch Diagonalen in lauter m-Ecke zerlegen laesst, mit Bezug auf einige Abhandlungen der Herren Lame, Rodrigues, Binet, Catalan und Duhamel in dem Journal de Mathematiques pures et appliquees, publie par Joseph Liouville. T. III. IV., Archiv der Mathematik u. Physik, 1 (1841), pp. 193ff; see especially p. 198.
B. Sury, Generalized Catalan numbers: linear recursion and divisibility, JIS 12 (2009), Article 09.7.5.
Lajos Takacs, Enumeration of rooted trees and forests, Math. Scientist 18 (1993), 1-10, esp. Eq. (5).
Jun Yan, Lattice paths enumerations weighted by ascent lengths, arXiv:2501.01152 [math.CO], 2025. See p. 7.

Cf. A001764, A002293, A002294, A002295.
Sixth column of triangle A062993.
Cf. A235535: binomial(9n,3n)/(6n+1); A235536: binomial(8n,2n)/(6n+1).
Cf. A258708.
Cf. A130564.

a002296 n = a002296_list !! n
a002296_list = [a258708 (4 * n) (3 * n) | n <- [1..]]
-- Reinhard Zumkeller, Jun 23 2015

seq(binomial(7*n+1, n)/(7*n+1), n=0..30); # Robert FERREOL, Apr 02 2015
n:=30: G:=series(RootOf(g = 1+x*g^7, g), x=0, n+1): seq(coeff(G, x, k), k=0..n); # Robert FERREOL, Apr 02 2015

Table[Binomial[7n,n]/(6n+1),{n,0,20}] (* Harvey P. Dale, Nov 21 2011 *)

a(n)=binomial(7*n,n)/(6*n+1) \\ Charles R Greathouse IV, Feb 06 2012

O.g.f. A(x) = 1 + x*A(x)^7 = 1/(1-x*A(x)^6).
a(n) = binomial(7*n,n-1)/n, n>=1, a(0)=1. From the Lagrange series of the o.g.f. A(x) with its above given implicit equation.
D-finite with recurrence: 72*n*(6*n-1)*(3*n-1)*(2*n-1)*(3*n-2)*(6*n+1)*a(n) - 7*(7*n-3)*(7*n-6)*(7*n-2)*(7*n-5)*(7*n-1)*(7*n-4)*a(n-1) = 0. - R. J. Mathar, Nov 16 2012
a(n) are special values of Jacobi polynomials, in Maple notation:
  a(n) = JacobiP(n-1, 6*n+1, -n, 1)/n, n = 1, 2, ... . - Karol A. Penson, Mar 16 2015
a(n) = binomial(7*n+1, n)/(7*n+1) = A062993(n+5,5). - Robert FERREOL, Apr 02 2015
a(0) = 1; a(n) = Sum_{i1+i2+...+i7=n-1} a(i1)*a(i2)*...*a(i7) for n>=1. - Robert FERREOL, Apr 02 2015
From Ilya Gutkovskiy, Jan 16 2017: (Start)
O.g.f.: 6F5(1/7,2/7,3/7,4/7,5/7,6/7; 1/3,1/2,2/3,5/6,7/6; 823543*x/46656).
E.g.f.: 6F6(1/7,2/7,3/7,4/7,5/7,6/7; 1/3,1/2,2/3,5/6,1,7/6; 823543*x/46656).
a(n) ~ 7^(7*n+1/2)/(sqrt(Pi)*3^(6*n+3/2)*4^(3*n+1)*n^(3/2)). (End)
x*A'(x)/A(x) = (A(x) - 1)/(- 6*A(x) + 7) = x + 13*x^2 + 190*x^3 + 2925*x^4 + ... = (1/7)*Sum_{n >= 1} binomial(7*n,n)*x^n. Cf. A001764 and A002293, A002294, A002295. - Peter Bala, Feb 04 2022
G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^13). - Seiichi Manyama, Jun 16 2025

Pfaff-Fuss-Catalan, Raney, o.g.f. and 7-ary tree comments from Wolfdieter Lang, Sep 14 2007

A000260 Number of rooted simplicial 3-polytopes with n+3 nodes; or rooted 3-connected triangulations with 2n+2 faces; or rooted 3-connected trivalent maps with 2n+2 vertices.

Original entry on oeis.org

1, 1, 3, 13, 68, 399, 2530, 16965, 118668, 857956, 6369883, 48336171, 373537388, 2931682810, 23317105140, 187606350645, 1524813969276, 12504654858828, 103367824774012, 860593023907540, 7211115497448720, 60776550501588855

Offset: 0

N. J. A. Sloane

Number of rooted loopless planar maps with n edges. E.g., there are a(2)=3 loopless planar maps with 2 edges: two rooted paths (.-.-.) and one digon (.=.). - Valery A. Liskovets, Sep 25 2003
Number of intervals (i.e., ordered pairs (x,y) such that x<=y) in the Tamari lattice (rotation lattice of binary trees) of size n (see Pallo and Chapoton references). - Ralf Stephan, May 08 2007, Jean Pallo (Jean.Pallo(AT)u-bourgogne.fr), Sep 11 2007
Number of rooted triangulations of type [n, 0] (see Brown paper eq (4.8)). - Michel Marcus, Jun 23 2013
Equivalently, number of rooted bridgeless planar maps with n edges. - Noam Zeilberger, Oct 06 2016
The September 2018 talk by Noam Zeilberger (see link to video) connects three topics (planar maps, Tamari lattices, lambda calculus) and eight sequences: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827. - N. J. A. Sloane, Sep 17 2018
Number of uniquely sorted permutations of [2n+1] that avoid the pattern 231. Also the number of uniquely sorted permutations of [2n+1] that avoid 132. - Colin Defant, Jun 13 2019
The sequence 1,3,13,68,... appears naturally in integral geometry, namely in the algebra of unitarily invariant valuations on complex space forms. - Andreas Bernig, Feb 02 2020

			G.f. = 1 + x + 3*x^2 + 13*x^3 + 68*x^4 + 399*x^5 + 2530*x^6 + 16965*x^7 + ...

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.
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 714.
Handbook of Combinatorics, North-Holland '95, p. 891.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
W. T. Tutte, The enumerative theory of planar maps, in A Survey of Combinatorial Theory (J. N. Srivastava et al. eds.), pp. 437-448, North-Holland, Amsterdam, 1973.

T. D. Noe, Table of n, a(n) for n = 0..200
E. A. Bender and N. C. Wormald, The number of loopless planar maps, Discr. Math. 54:2 (1985), 235-237.
Andreas Bernig, Unitarily invariant valuations and Tutte's sequence, arXiv:2001.03372 [math.DG], 2020.
Alin Bostan, Frédéric Chyzak, Bérénice Delcroix-Oger, Guillaume Laplante-Anfossi, Vincent Pilaud, and Kurt Stoeckl, Diagonals of permutahedra and associahedra, Sém. Lotharingien Comb., 37th Formal Power Series Alg. Comb. (FPSAC 2025). See pp. 10-11.
Alin Bostan, Frédéric Chyzak, and Vincent Pilaud, Refined product formulas for Tamari intervals, arXiv:2303.10986 [math.CO], 2023.
T. Daniel Brennan, Christian Ferko, and Savdeep Sethi, A Non-Abelian Analogue of DBI from $T \overline{T}$, arXiv:1912.12389 [hep-th], 2019. See also SciPost Phys. (2020) Vol. 8, 052.
Enrica Duchi and Corentin Henriet, A bijection between Tamari intervals and extended fighting fish, arXiv:2206.04375 [math.CO], 2022.
William G. Brown, Enumeration of Triangulations of the Disk, Proc. Lond. Math. Soc. s3-14 (1964) 746-768.
Frédéric Chapoton, Sur le nombre d'intervalles dans les treillis de Tamari, arXiv:math/0602368 [math.CO], 2006.
Grégory Chatel, Vincent Pilaud, and Viviane Pons, The weak order on integer posets, arXiv:1701.07995 [math.CO], 2017.
Grégory Chatel and Viviane Pons, Counting smaller elements in the Tamari and m-Tamari lattices, arXiv preprint arXiv:1311.3922 [math.CO], 2013.
François David, A model of random surfaces with nontrivial critical behavior, Nuclear Physics B, v. 257 (1985), 543-576.
Colin Defant, Catalan Intervals and Uniquely Sorted Permutations, arXiv:1904.02627 [math.CO], 2019.
C. F. Earl and L. J. March, Architectural applications of graph theory, pp. 327-355 of R. J. Wilson and L. W. Beineke, editors, Applications of Graph Theory. Academic Press, NY, 1979. (Annotated scanned copy)
Wenjie Fang, Planar triangulations, bridgeless planar maps and Tamari intervals, arXiv:1611.07922 [math.CO], 2016.
Joël Gay and Vincent Pilaud, The weak order on Weyl posets, arXiv:1804.06572 [math.CO], 2018.
C. Germain and J. Pallo, The number of coverings in four Catalan lattices, Intern. J. Computer Math., Vol. 61 (1996) pp. 19-28. (See p. 27.)
Hsien-Kuei Hwang, Mihyun Kang, and Guan-Huei Duh, Asymptotic Expansions for Sub-Critical Lagrangean Forms, LIPIcs Proceedings of Analysis of Algorithms 2018, Vol. 110. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2018.
Zhaoxiang Li and Yanpei Liu, Chromatic sums of general maps on the sphere and the projective plane, Discr. Math. 307 (2007), 78-87.
Michaël Moortgat, The Tamari order for D^3 and derivability in semi-associative Lambek-Grishin Calculus, 15th Workshop: Computational Logic and Applications (CLA 2020).
K. A. Penson, K. Górska, A. Horzela, and G. H. E. Duchamp, Hausdorff moment problem for combinatorial numbers of Brown and Tutte: exact solution, arXiv:2209.06574 [math.CO], 2022.
Viviane Pons, Combinatorics of the Permutahedra, Associahedra, and Friends, arXiv:2310.12687 [math.CO], 2023. See p. 37.
W. T. Tutte, A census of Hamiltonian polygons, Canad. J. Math., 14 (1962), 402-417.
William T. Tutte, A census of planar triangulations, Canad. J. Math. 14 (1962), 21-38. See Eq. 5.12.
William T. Tutte, On the enumeration of convex polyhedra, J. Combin. Theory Ser. B 28 (1980), no. 2, 105-126. MR0572468 (81j:05073).
T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus. III: Nonseparable maps, J. Combinatorial Theory Ser. B 18 (1975), 222-259, eq. (6).
Noam Zeilberger, A sequent calculus for the Tamari order, arXiv:1701.02917 [cs.LO], 2017.
Noam Zeilberger, A Sequent Calculus for a Semi-Associative Law, arXiv preprint 1803.10030 [math.LO], 2018-2019 (A revised version of a 2017 conference paper).
Noam Zeilberger, A theory of linear typings as flows on 3-valent graphs, arXiv:1804.10540 [cs.LO], 2018.
Noam Zeilberger, A proof-theoretic analysis of the rotation lattice of binary trees, Part 1 (video), Rutgers Experimental Math Seminar, Sep 13 2018. See also Part 2.

Row sums of A342981.
Column 0 of A146305 and A341856; Column 2 of A255918.
Sequences mentioned in the Noam Zeilberger 2018 video: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827.
Cf. A000256, A006013, A027836, A069271, A263191, A290326.

[Binomial(4*n+1, n+1)-9*Binomial(4*n+1, n-1): n in [0..25]]; // Vincenzo Librandi, Nov 24 2016

A000260 := proc(n)
    2*(4*n+1)!/((n+1)!*(3*n+2)!) ;
end proc:

Table[Binomial[4n+1,n+1]-9*Binomial[4n+1,n-1],{n,0,25}] (* Harvey P. Dale, Aug 23 2011 *)
a[ n_] := SeriesCoefficient[ HypergeometricPFQ[ {1/2, 3/4, 1, 5/4}, {4/3, 5/3, 2}, 256/27 x], {x, 0, n}]; (* Michael Somos, Dec 23 2014 *)
terms = 22; G[] = 0; Do[G[x] = 1+x*G[x]^4 + O[x]^terms, terms];
CoefficientList[(2-G[x])*G[x]^2, x] (* Jean-François Alcover, Jan 13 2018, after Mark van Hoeij *)

{a(n) = if( n<0, 0, 2 * (4*n + 1)! / ((n + 1)! * (3*n + 2)!))}; /* Michael Somos, Sep 07 2005 */

{a(n) = binomial( 4*n + 2, n + 1) / ((2*n + 1) * (3*n + 2))}; /* Michael Somos, Mar 28 2012 */

def a(n):
    n = ZZ(n)
    return (4*n + 2).binomial(n + 1) // ((2*n + 1) * (3*n + 2))
# F. Chapoton, Aug 06 2015

a(n) = 2*(4*n+1)! / ((n+1)!*(3*n+2)!) = binomial(4*n+1, n+1) - 9*binomial(4*n+1, n-1).
G.f.: (2-g)*g^2 where g = 1 + x*g^4 is the g.f. of A002293. - Mark van Hoeij, Nov 10 2011
G.f.: hypergeom([1,1/2,3/4,5/4],[2,4/3,5/3],256*x/27) = 1 + 120*x/(Q(0)-120*x); Q(k) = 8*x*(2*k+1)*(4*k+3)*(4*k+5) + 3*(k+2)*(3*k+4)*(3*k+5) - 24*x*(k+2)*(2*k+3)*(3*k+4)*(3*k+5)*(4*k+7)*(4*k+9)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 25 2011
a(n) = binomial(4*n + 2, n + 1) / ((2*n + 1) * (3*n + 2)). - Michael Somos, Mar 28 2012
a(n) * (n+1) = A069271(n). - Michael Somos, Mar 28 2012
0 = F(a(n), a(n+1), ..., a(n+8)) for all n in Z where a(-1) = 3/4 and F() is a polynomial of degree 2 with integer coefficients and 29 monomials. - Michael Somos, Dec 23 2014
D-finite with recurrence: 3*(3*n+2)*(3*n+1)*(n+1)*a(n) - 8*(4*n+1)*(2*n-1)*(4*n-1)*a(n-1) = 0. - R. J. Mathar, Oct 21 2015
a(n) = Sum_{k=1..A000108(n)} k * A263191(n,k). - Alois P. Heinz, Nov 16 2015
a(n) ~ 2^(8*n+7/2) / (sqrt(Pi) * n^(5/2) * 3^(3*n+5/2)). - Vaclav Kotesovec, Feb 26 2016
E.g.f.: 3F3(1/2,3/4,5/4; 4/3,5/3,2; 256*x/27). - Ilya Gutkovskiy, Feb 01 2017
From Gheorghe Coserea, Aug 17 2017: (Start)
G.f. y(x) satisfies:
0 = x^3*y^4 + 3*x^2*y^3 + x*(8*x+3)*y^2 - (20*x-1)*y + 16*x-1.
0 = x*(256*x - 27)*deriv(y,x) - 8*x^2*y^3 - 25*x*y^2 + 4*(24*x-11)*y + 44.
(End)
From Karol A. Penson, Apr 06 2022: (Start)
a(n) = Integral_{x=0...256/27} x^n*W(x), where
W(x) = (sqrt(2)/Pi)*(h1(x) - h2(x) + h3(x)) and
h1(x) = 3F2([-6/12,-2/12,  2/12], [ 3/12,  9/12], (27*x)/256)/((x/2)^(1/2));
h2(x) = 3F2([-3/12, 1/12,  5/12], [ 6/12, 15/12], (27*x)/256)/(x^(1/4));
h3(x) = 3F2([ 3/12, 7/12, 11/12], [18/12, 21/12], (27*x)/256)/(x^(-1/4)*32).
This integral representation is unique as the solution of n-th Hausdorff power moment of the function W. Using only the definition of a(n), W(x) can be proven to be positive. W(x) is singular at x = 0 and for x > 0 is monotonically decreasing to zero at x = 256/27. (End)
a(n) = (1/27^n) * Product_{1 <= i <= j <= 3*n} (3*i + j + 3)/(3*i + j - 1). Cf. A006013. - Peter Bala, Feb 21 2023

Edited by F. Chapoton, Feb 03 2011

cp's OEIS Frontend

A381942 G.f. A(x) satisfies A(x) = (1 + x)^3 * B(x*A(x)), where B(x) is the g.f. of A002293.

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A381945 G.f. A(x) satisfies A(x) = B(x*A(x)) / (1 - x)^2, where B(x) is the g.f. of A002293.

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A381947 G.f. A(x) satisfies A(x) = B(x*A(x)) / (1 - x)^3, where B(x) is the g.f. of A002293.

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A380638 Expansion of e.g.f. exp(x*G(4*x)^4) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

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

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A002294 a(n) = binomial(5*n, n)/(4*n + 1).

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A002295 Number of dissections of a polygon: binomial(6n,n)/(5n+1).

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A380638 Expansion of e.g.f. exp(xG(4x)^4) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

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

A002294 a(n) = binomial(5n, n)/(4n + 1).