A005676 -id:A005676 - 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

A002293 Number of dissections of a polygon: binomial(4n, n)/(3n + 1).

Original entry on oeis.org

1, 1, 4, 22, 140, 969, 7084, 53820, 420732, 3362260, 27343888, 225568798, 1882933364, 15875338990, 134993766600, 1156393243320, 9969937491420, 86445222719724, 753310723010608, 6594154339031800, 57956002331347120, 511238042454541545

Offset: 0

N. J. A. Sloane

The number of rooted loopless n-edge maps in the plane (planar with a distinguished outside face). - Valery A. Liskovets, Mar 17 2005
Number of lattice paths from (1,0) to (3*n+1,n) which, starting from (1,0), only utilize the steps +(1,0) and +(0,1) and additionally, the paths lie completely below the line y = (1/3)*x (i.e., if (a,b) is in the path, then b < a/3). - Joseph Cooper (jecooper(AT)mit.edu), Feb 07 2006
Number of length-n restricted growth strings (RGS) [s(0), s(1), ..., s(n-1)] where s(0) = 0 and s(k) <= s(k-1) + 3, see fxtbook link below. - Joerg Arndt, Apr 08 2011
From Wolfdieter Lang, Sep 14 2007: (Start)
a(n), n >= 1, enumerates quartic trees (rooted, ordered, incomplete) with n vertices (including the root).
Pfaff-Fuss-Catalan sequence C^{m}_n for m = 4. See the Graham et al. reference, p. 347. eq. 7.66. (Second edition, p. 361, eq. 7.67.) See also the Pólya-Szegő reference.
Also 4-Raney sequence. See the Graham et al. reference, pp. 346-347.
(End)
Bacher: "We describe the statistics of checkerboard triangulations obtained by coloring black every other triangle in triangulations of convex polygons." The current sequence (A002293) occurs on p. 12 as one of two "extremal sequences" of an array of coefficients of polynomials, whose generating functions are given in terms of hypergeometric functions. - Jonathan Vos Post, Oct 05 2007
A generating function in terms of a (labyrinthine) solution to a depressed quartic equation is given in the Copeland link for signed A005810. With D(z,t) that g.f., a g.f. for signed A002293 is {[-1+1/D(z,t)]/(4t)}^(1/3). - Tom Copeland, Oct 10 2012
For a relation to the inviscid Burgers's equation, see A001764. - Tom Copeland, Feb 15 2014
For relations to compositional inversion, the Legendre transform, and convex geometry, see the Copeland, the Schuetz and Whieldon, and the Gross (p. 58) links. - Tom Copeland, Feb 21 2017 (See also Gross et al. in A062994. - Tom Copeland, Dec 24 2019)
This is the number of A'Campo bicolored forests of degree n and co-dimension 0. This can be shown using generating functions or a combinatorial approach. See Combe and Jugé link below. - Noemie Combe, Feb 28 2017
Conjecturally, a(n) is the number of 3-uniform words over the alphabet [n] that avoid the patterns 231 and 221 (see the Defant and Kravitz link). - Colin Defant, Sep 26 2018
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. Cf. A001764. - Tom Copeland, Dec 13 2019
a(n) is the total number of down steps before the first up step in all 3_1-Dyck paths of length 4*n. A 3_1-Dyck path is a lattice path with steps (1, 3), (1, -1) that starts and ends at y = 0 and stays above the line y = -1. - Sarah Selkirk, May 10 2020
a(n) is the number of pairs (A<=B) of noncrossing partitions of [2n] such that every block of A has exactly two elements. In fact, it is proved that a(n) is the number of planar tied arc diagrams with n arcs (see Aicardi link below). A planar diagram with n arcs represents a noncrossing partition A of [2n] with n blocks, each block containing the endpoints of one arc; each tie connects two arcs, so that the ties define a partition B >= A: the endpoints of two arcs connected by a tie belong to the same block of B. Ties do not cross arcs nor other ties iff B has a planar diagram, i.e., B is a noncrossing partition. - Francesca Aicardi, Nov 07 2022
Dropping the initial 1 (starting 1, 4, 22 with offset 1) yields the REVERT transformation 1, -4 ,10, -20, 35.. essentially A000292 without leading 0. - R. J. Mathar, Aug 17 2023
Number of rooted polyominoes composed of n pentagonal cells of the hyperbolic regular tiling with Schläfli symbol {5,oo}. A rooted polyomino has one external edge identified, and chiral pairs are counted as two. A stereographic projection of the {5,oo} tiling on the Poincaré disk can be obtained via the Christensson link. - Robert A. Russell, Jan 27 2024
This is instance k = 4 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
a(n) is the cardinality of the planar ramified Jones monoid PR(J_n). - Diego Arcis, Nov 21 2024

			There are a(2) = 4 quartic trees (vertex degree <= 4 and 4 possible branchings) with 2 vertices (one of them the root). Adding one more branch (one more vertex) to these four trees yields 4*4 + 6 = 22 = a(3) such trees.

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Column k=3 of triangle A062993 and A070914.
Cf. A000260, A002295, A002296, A027836, A062994, A346646 (binomial transform), A346664 (inverse binomial transform).
Cf. A006632, A006633, A006634, A025174, A069271, A196678, A224274, A233658, A233666, A233667, A277877, A283049, A283101, A283102, A283103.
Polyominoes: A005038 (oriented), A005040 (unoriented), A369471 (chiral), A369472 (achiral), A001764 {4,oo}, A002294 {6,oo}.
Cf. A130564 (for generalized Catalan C(k, n), for = 4).

List([0..22],n->Binomial(4*n,n)/(3*n+1)); # Muniru A Asiru, Nov 01 2018

[ Binomial(4*n,n)/(3*n+1): n in [0..50]]; // Vincenzo Librandi, Apr 19 2011

series(RootOf(g = 1+x*g^4, g),x=0,20); # Mark van Hoeij, Nov 10 2011
seq(binomial(4*n, n)/(3*n+1),n=0..20); # Robert FERREOL, Apr 02 2015
# Using the integral representation above:
Digits:=6;
R:=proc(x)((I + sqrt(3))*(4*sqrt(256 - 27*x) - 12*I*sqrt(3)*sqrt(x))^(1/3))/16 - ((I - sqrt(3))*(4*sqrt(256 - 27*x) + 12*I*sqrt(3)*sqrt(x))^(1/3))/16;end;
W:=proc(x) x^(-3/4) * sqrt(4*R(x) - 3^(3/4)*x^(1/4)/sqrt(R(x)))/(2*3^(1/4)*Pi);end;
# Attention: W(x) is singular at x = 0. Integration is done from  a very small positive x to x = 256/27.
# For a(8):  ... gives 420732
evalf(int(x^8*W(x),x=0.000001..256/27));
# Karol A. Penson, Jul 05 2024

CoefficientList[InverseSeries[ Series[ y - y^4, {y, 0, 60}], x], x][[Range[2, 60, 3]]]
Table[Binomial[4n,n]/(3n+1),{n,0,25}] (* Harvey P. Dale, Apr 18 2011 *)
CoefficientList[1 + InverseSeries[Series[x/(1 + x)^4, {x, 0, 60}]], x] (* Gheorghe Coserea, Aug 12 2015 *)
terms = 22; A[] = 0; Do[A[x] = 1 + x*A[x]^4 + O[x]^terms, terms];
CoefficientList[A[x], x] (* Jean-François Alcover, Jan 13 2018 *)

a(n)=binomial(4*n,n)/(3*n+1) /* Charles R Greathouse IV, Jun 16 2011 */

my(x='x+O('x^33)); Vec(1 + serreverse(x/(1+x)^4)) \\ Gheorghe Coserea, Aug 12 2015

A002293_list, x = [1], 1
for n in range(100):
    x = x*4*(4*n+3)*(4*n+2)*(4*n+1)//((3*n+2)*(3*n+3)*(3*n+4))
    A002293_list.append(x) # Chai Wah Wu, Feb 19 2016

O.g.f. satisfies: A(x) = 1 + x*A(x)^4 = 1/(1 - x*A(x)^3).
a(n) = binomial(4*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.
From Karol A. Penson, Apr 02 2010: (Start)
Integral representation as n-th Hausdorff power moment of a positive function on the interval [0, 256/27]:
a(n) = Integral_{x=0..256/27}(x^n((3/256) * sqrt(2) * sqrt(3) * ((2/27) * 3^(3/4) * 27^(1/4) * 256^(/4) * hypergeom([-1/12, 1/4, 7/12], [1/2, 3/4], (27/256)*x)/(sqrt(Pi) * x^(3/4)) - (2/27) * sqrt(2) * sqrt(27) * sqrt(256) * hypergeom([1/6, 1/2, 5/6], [3/4, 5/4], (27/256)*x)/ (sqrt(Pi) * sqrt(x)) - (1/81) * 3^(1/4) * 27^(3/4) * 256^(1/4) * hypergeom([5/12, 3/4, 13/12], [5/4, 3/2], (27/256)*x/(sqrt(Pi)*x^(1/4)))/sqrt(Pi))).
This representation is unique as it represents the solution of the Hausdorff moment problem.
O.g.f.: hypergeom([1/4, 1/2, 3/4], [2/3, 4/3], (256/27)*x);
E.g.f.: hypergeom([1/4, 1/2, 3/4], [2/3, 1, 4/3], (256/27)*x). (End)
a(n) = upper left term in M^n, M = the production matrix:
  1, 1
  3, 3, 1
  6, 6, 3, 1
  ...
(where 1, 3, 6, 10, ...) is the triangular series. - Gary W. Adamson, Jul 08 2011
O.g.f. satisfies g = 1+x*g^4. If h is the series reversion of x*g, so h(x*g)=x, then (x-h(x))/x^2 is the o.g.f. of A006013. - Mark van Hoeij, Nov 10 2011
a(n) = binomial(4*n+1, n)/(4*n+1) = A062993(n+2,2). - Robert FERREOL, Apr 02 2015
a(n) = Sum_{i=0..n-1} Sum_{j=0..n-1-i} Sum_{k=0..n-1-i-j} a(i)*a(j)*a(k)*a(n-1-i-j-k) for n>=1; and a(0) = 1. - Robert FERREOL, Apr 02 2015
a(n) ~ 2^(8*n+1/2) / (sqrt(Pi) * n^(3/2) * 3^(3*n+3/2)). - Vaclav Kotesovec, Jun 03 2015
From Peter Bala, Oct 16 2015: (Start)
A(x)^2 is o.g.f. for A069271; A(x)^3 is o.g.f. for A006632;
A(x)^5 is o.g.f. for A196678; A(x)^6 is o.g.f. for A006633;
A(x)^7 is o.g.f. for A233658; A(x)^8 is o.g.f. for A233666;
A(x)^9 is o.g.f. for A006634; A(x)^10 is o.g.f. for A233667. (End)
D-finite with recurrence: a(n+1) = a(n)*4*(4*n + 3)*(4*n + 2)*(4*n + 1)/((3*n + 2)*(3*n + 3)*(3*n + 4)). - Chai Wah Wu, Feb 19 2016
E.g.f.: F([1/4, 1/2, 3/4], [2/3, 1, 4/3], 256*x/27), where F is the generalized hypergeometric function. - Stefano Spezia, Dec 27 2019
x*A'(x)/A(x) = (A(x) - 1)/(- 3*A(x) + 4) = x + 7*x^2 + 55*x^3 + 455*x^4 + ... is the o.g.f. of A224274. Cf. A001764 and A002294 - A002296. - Peter Bala, Feb 04 2022
a(n) = hypergeom([1 - n, -3*n], [2], 1). Row sums of A173020. - Peter Bala, Aug 31 2023
G.f.: t*exp(4*t*hypergeom([1, 1, 5/4, 3/2, 7/4], [4/3, 5/3, 2, 2], (256*t)/27))+1. - Karol A. Penson, Dec 20 2023
From Karol A. Penson, Jul 03 2024: (Start)
a(n) = Integral_{x=0..256/27} x^(n)*W(x)dx, n>=0, where W(x) = x^(-3/4) * sqrt(4*R(x) - 3^(3/4)*x^(1/4)/sqrt(R(x)))/(2*3^(1/4)*Pi), with R(x) = ((i + sqrt(3))*(4*sqrt(256 - 27*x) -12*i*sqrt(3*x))^(1/3))/16 - ((i - sqrt(3))*(4*sqrt(256 - 27*x) + 12*i* sqrt(3*x))^(1/3))/16, where i is the imaginary unit.
The elementary function W(x) is positive on the interval x = (0, 256/27) and is equal to the combination of hypergeometric functions in my formula from 2010; see above.
(Pi*W(x))^6 satisfies an algebraic equation of order 6, with integer polynomials as coefficients. (End)
G.f.: (Sum_{n >= 0} binomial(4*n+1, n)*x^n) / (Sum_{n >= 0} binomial(4*n, n)*x^n). - Peter Bala, Dec 14 2024
G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^7). - Seiichi Manyama, Jun 16 2025

A005251 a(0) = 0, a(1) = a(2) = a(3) = 1; thereafter, a(n) = a(n-1) + a(n-2) + a(n-4).

Original entry on oeis.org

0, 1, 1, 1, 2, 4, 7, 12, 21, 37, 65, 114, 200, 351, 616, 1081, 1897, 3329, 5842, 10252, 17991, 31572, 55405, 97229, 170625, 299426, 525456, 922111, 1618192, 2839729, 4983377, 8745217, 15346786, 26931732, 47261895, 82938844, 145547525, 255418101, 448227521

Offset: 0

N. J. A. Sloane and R. K. Guy

a(n+3) is the number of n-bit sequences that avoid 010. Example: For n=4 the 12 sequences are all 4-bit sequences except 0100, 0101, 0010, 1010. - David Callan, Mar 25 2004
a(n+2) is the number of compositions (ordered partitions) of n where no two adjacent parts are != 1, see example. - Joerg Arndt, Jan 26 2013
a(n+1) is the number of compositions of n avoiding the part 2. - Joerg Arndt, Jul 13 2014
Number of different positive braids with n crossings of 3 strands.
This is a_2(n) in the Doroslovacki reference. Note that there is a typo in the paper in the formula for a_2(n): the upper bound in the inner sum should be "n-i" not "i-1". - Max Alekseyev, Jun 26 2007
a(n) is the number of peakless Motzkin paths of length n-1 with no UHH...HD's starting at level > 0 (here n > 0 and U=(1,1), H=(1,0), D=(1,-1)). Example: a(5)=7 because from all 8 peakless Motzkin paths of length 5 (see A004148) only UUHDD does not qualify. - Emeric Deutsch, Sep 13 2004
Equals the INVERT transform of (1, 0, 1, 1, 1, ...); equivalent to a(n) = a(n-1) + a(n-3) + a(n-4) + ... - Gary W. Adamson, Apr 27 2009
a(n) is the number of length n-1 words on {0,1} such that each string of 1's is followed by a string of at least two 0's. For example, a(5) = 4 because we have: 0000, 0100, 1000, and 1100. - Geoffrey Critzer, Aug 09 2013
a(n+1) is the top left entry of the n-th power of any of the 3 X 3 matrices [1, 1, 0; 0, 1, 1; 1, 0, 0] or [1, 0, 1; 1, 1, 0; 0, 1, 0] or [1, 1, 0; 0, 0, 1; 1, 0, 1] or [1, 0, 1; 1, 0, 0; 0, 1, 1]. - R. J. Mathar, Feb 03 2014
For n >= 2, a(n) is the number of (n-2)-length binary words with no isolated zeros. - Milan Janjic, Mar 07 2015
Apart from the first three terms, the total number of bargraphs of semiperimeter n of height at most two for n >= 2 starts 1, 2, 4, 7, 12, ... - Arnold Knopfmacher, Nov 02 2016
Number of DD-equivalence classes of Łukasiewicz paths. Łukasiewicz paths are DD-equivalent iff the positions of pattern DD are identical in these paths. - Sergey Kirgizov, Apr 08 2018
From Gus Wiseman, Nov 25 2019: (Start)
For n > 0, also the number of subsets of {1, ..., n - 3} such that if x and x + 2 are both in the subset, then so is x + 1. For example, the a(3) = 1 through a(7) = 12 subsets are:
  {}  {}   {}     {}       {}
      {1}  {1}    {1}      {1}
           {2}    {2}      {2}
           {1,2}  {3}      {3}
                  {1,2}    {4}
                  {2,3}    {1,2}
                  {1,2,3}  {1,4}
                           {2,3}
                           {3,4}
                           {1,2,3}
                           {2,3,4}
                           {1,2,3,4}
(End)
The two-dimensional version, which counts sets of pairs where, if two pairs are separated by graph-distance 2, then the intermediate pair or pairs are also in the set, is A329871. - Gus Wiseman, Nov 30 2019
a(n+1) is the number of ways to tile a strip of length n with squares, dominoes, and tetrominoes, where the first tile cannot be a domino. - Greg Dresden and Myanna Nash, Aug 18 2020
For n>=3, a(n) is the number of binary strings of length n-2 without any maximal runs of ones of length 1. - Félix Balado, Aug 25 2025

			From _Joerg Arndt_, Jan 26 2013: (Start)
The a(5+2) = 12 compositions of 5 where no two adjacent parts are != 1 are
  [ 1]  [ 1 1 1 1 1 ]
  [ 2]  [ 1 1 1 2 ]
  [ 3]  [ 1 1 2 1 ]
  [ 4]  [ 1 1 3 ]
  [ 5]  [ 1 2 1 1 ]
  [ 6]  [ 1 3 1 ]
  [ 7]  [ 1 4 ]
  [ 8]  [ 2 1 1 1 ]
  [ 9]  [ 2 1 2 ]
  [10]  [ 3 1 1 ]
  [11]  [ 4 1 ]
  [12]  [ 5 ]
(End)
G.f. = x + x^2 + x^3 + 2*x^4 + 4*x^5 + 7*x^6 + 12*x^7 + 21*x^8 + 37*x^9 + ...

S. Burckel, Efficient methods for three strand braids (submitted). [Apparently unpublished]
P. Chinn and S. Heubach, "Compositions of n with no occurrence of k", Congressus Numeratium, 2002, v. 162, pp. 33-51.
John H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, p. 205.
R. K. Guy, "Anyone for Twopins?" in D. A. Klarner, editor, The Mathematical Gardner. Prindle, Weber and Schmidt, Boston, 1981, pp. 2-15.
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..500
Kassie Archer, Ethan Borsh, Jensen Bridges, Christina Graves, and Millie Jeske, Cyclic permutations avoiding patterns in both one-line and cycle forms, arXiv:2312.05145 [math.CO], 2023. See p. 2.
Andrei Asinowski and Cyril Banderier, On Lattice Paths with Marked Patterns: Generating Functions and Multivariate Gaussian Distribution, 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020) Leibniz International Proceedings in Informatics (LIPIcs) Vol. 159, 1:1-1:16.
R. Austin and R. K. Guy, Binary sequences without isolated ones, Fib. Quart., 16 (1978), 84-86.
J.-L. Baril, Avoiding patterns in irreducible permutations, Discrete Mathematics and Theoretical Computer Science, Vol 17, No 3 (2016). See Table 4.
Jean-Luc Baril, Sergey Kirgizov and Armen Petrossian, Enumeration of Łukasiewicz paths modulo some patterns, arXiv:1804.01293 [math.CO], 2018.
N. Bergeron, S. Mykytiuk, F. Sottile and S. van Willigenburg, Shifted quasisymmetric functions and the Hopf algebra of peak functions, arXiv:math/9904105 [math.CO], 1999.
D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, Example 11.
A. Blecher, C. Brennan, A. Knopfmacher and H. Prodinger, The height and width of bargraphs, Discrete Applied Math. 180, (2015), 36-44.
A. Brousseau, Fibonacci and Related Number Theoretic Tables, Fibonacci Association, San Jose, CA, 1972, p. 112.
P. Chinn and S. Heubach, Integer Sequences Related to Compositions without 2's, J. Integer Seqs., Vol. 6, 2003.
James Currie, Pascal Ochem, Narad Rampersad, and Jeffrey Shallit, Properties of a Ternary Infinite Word, arXiv:2206.01776 [cs.DM], 2022.
James Currie, Pascal Ochem, Narad Rampersad, and Jeffrey Shallit, Complement Avoidance in Binary Words, arXiv:2209.09598 [math.CO], 2022.
J. Demetrovics et al., On the number of unions in a family of sets, in Combinatorial Math., Proc. 3rd Internat. Conf., Annals NY Acad. Sci., 555 (1989), 150-158.
R. Doroslovacki, Binary sequences without 011...110 (k-1 1's) for fixed k, Mat. Vesnik 46 (1994), no. 3-4, 93-98.
Nazim Fatès, Biswanath Sethi, and Sukanta Das, On the Reversibility of ECAs with Fully Asynchronous Updating: The Recurrence Point of View, in Reversibility and Universality, Andrew Adamatzky, editor, Emergence, Complexity and Computation Vol. 30. Springer, 2018.
Steven Finch, Cantor-solus and Cantor-multus distributions, arXiv:2003.09458 [math.CO], 2020.
R. L. Graham and N. J. A. Sloane, Anti-Hadamard matrices, Linear Alg. Applic., 62 (1984), 113-137.
R. K. Guy, Letter to N. J. A. Sloane, Feb 1986
R. K. Guy, Anyone for Twopins?, in D. A. Klarner, editor, The Mathematical Gardner. Prindle, Weber and Schmidt, Boston, 1981, pp. 2-15. [Annotated scanned copy, with permission]
V. C. Harris and C. C. Styles, A generalization of Fibonacci numbers, Fib. Quart. 2 (1964) 277-289, sequence u(n,1,2).
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 98
Milan Janjic, Binomial Coefficients and Enumeration of Restricted Words, Journal of Integer Sequences, 2016, Vol 19, #16.7.3.
Vedran Krcadinac, A new generalization of the golden ratio, Fibonacci Quart. 44 (2006), no. 4, 335-340.
Erkko Lehtonen and Tamás Waldhauser, Associative spectra of graph algebras II. Satisfaction of bracketing identities, spectrum dichotomy, arXiv:2011.08522 [math.CO], 2020.
J. J. Madden, A generating function for the distribution of runs in binary words, arXiv:1707.04351 [math.CO], 2017, Theorem 1.1, r=2, k=0.
T. Mansour and M. Shattuck, Counting Peaks and Valleys in a Partition of a Set, J. Int. Seq. 13 (2010), 10.6.8, Lemma 2.1, k=2, 0 peaks.
Denis Neiter and Amsha Proag, Links Between Sums Over Paths in Bernoulli's Triangles and the Fibonacci Numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.8.3.
Nicolas Ollinger and Jeffrey Shallit, The Repetition Threshold for Rote Sequences, arXiv:2406.17867 [math.CO], 2024.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
A. G. Shannon, Some recurrence relations for binary sequence matrices, NNTDM 17 (2011), 4, 913.
Bojan Vučković and Miodrag Živković, Row Space Cardinalities Above 2^(n - 2) + 2^(n - 3), ResearchGate, January 2017, p. 3.
Index entries for linear recurrences with constant coefficients, signature (2,-1,1).

Cf. A001608, A004148, A005314, A006498, A011973, A049864, A049853, A078065, A118891, A173022, A176971, A178470, A261041, A303696, A329871, A384153.
Bisection of Padovan sequence A000931.
Partial sums of A005314 shifted 3 times to the right, if we assume A005314(0) = 1.
Compositions without adjacent equal parts are A003242.
Compositions without isolated parts are A114901.
Row sums of A097230(n-2) for n>1.

a005251 n = a005251_list !! n
a005251_list = 0 : 1 : 1 : 1 : zipWith (+) a005251_list
   (drop 2 $ zipWith (+) a005251_list (tail a005251_list))
-- Reinhard Zumkeller, Dec 28 2011

I:=[0,1,1,1]; [n le 4 select I[n] else Self(n-1)+Self(n-2)+Self(n-4): n in [1..45]]; // Vincenzo Librandi, Nov 30 2018

R:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x*(1-x)/(1-2*x + x^2 - x^3) )); // Marius A. Burtea, Oct 24 2019

A005251 := proc(n) option remember; if n <= 2 then n elif n = 3 then 4 else 2*A005251(n - 1) - A005251(n - 2) + A005251(n - 3); fi; end;
A005251:=(-1+z)/(-1+2*z-z**2+z**3); # Simon Plouffe in his 1992 dissertation
a := n -> `if`(n<=1, n, hypergeom([(2-n)/3, 1-n/3, (1-n)/3], [1/2, -n+1], 27/4)):
seq(simplify(a(n)), n=0..36); # Peter Luschny, Apr 08 2018

LinearRecurrence[{2,-1,1},{0,1,1},40]  (* Harvey P. Dale, May 05 2011 *)
a[ n_]:= If[n<0, SeriesCoefficient[ -x(1-x)/(1 -x + 2x^2 -x^3), {x, 0, -n}], SeriesCoefficient[ x(1-x)/(1 -2x +x^2 -x^3), {x, 0, n}]] (* Michael Somos, Dec 13 2013 *)
a[0] = 1; a[1] = a[2] = 0; a[n_] := a[n] = a[n-2] + a[n-3]; Table[a[2 n-1], {n, 1, 20}] (* Rigoberto Florez, Oct 15 2019 *)
Table[If[n==0,0,Length[DeleteCases[Subsets[Range[n-3]],{_,x_,y_,_}/;x+2==y]]],{n,0,10}] (* Gus Wiseman, Nov 25 2019 *)

Vec((1-x)/(1-2*x+x^2-x^3)+O(x^99)) /* Charles R Greathouse IV, Nov 20 2012 */

{a(n) = if( n<0, polcoeff( -x*(1-x)/(1 -x +2*x^2 -x^3) + x*O(x^-n), -n), polcoeff( x*(1-x)/(1 -2*x +x^2 -x^3) + x*O(x^n), n))} /* Michael Somos, Dec 13 2013 */

[sum( binomial(n-j-1, 2*j) for j in (0..floor((n-1)/3)) ) for n in (0..50)] # G. C. Greubel, Apr 13 2022

a(n) = 2*a(n-1) - a(n-2) + a(n-3).
G.f.: z*(1-z)/(1 - 2*z + z^2 - z^3). - Emeric Deutsch, Sep 13 2004
23*a_n = 3*P_{2n+1} + 7*P_{2n} - 2*P_{2n-1}, where P_n are the Perrin numbers, A001608. - Don Knuth, Dec 09 2008
a(n+1) = Sum_{k=0..n} binomial(n-k, 2k). - Richard L. Ollerton, May 12 2004
From Henry Bottomley, Feb 21 2001: (Start)
a(n) = (Sum_{ja(n) = A005314(n) - A005314(n-1).
a(n) = A049853(n-1) - a(n-1).
a(n) = A005314(n) - a(n-2). (End)
Conjecture: a(n+1) + |A078065(n)| = 2*A005314(n+1). - Creighton Dement, Dec 21 2004
a(n+2) has g.f. (F_3(-x) + F_2(-x))/(F_4(-x) + F_3(-x)) = 1/(-x+1/(-x+1/(-x+1))) where F_n(x) is the n-th Fibonacci polynomial; see A011973. - Qiaochu Yuan (qchu(AT)mit.edu), Feb 19 2009
a(n) = A173022(2^(n-2) - 1) for n > 1. - Reinhard Zumkeller, Feb 07 2010
BINOMIAL transform of A176971 is a(n+1). - Michael Somos, Dec 13 2013
a(n) = hypergeom([(2-n)/3, 1-n/3, (1-n)/3], [1/2, -n+1], 27/4) for n > 1. - Peter Luschny, Apr 08 2018
G.f.: z/(1-z-z^3-z^4-z^5-...) for the compositions of n-1 avoiding 2. The g.f. for the number of compositions of n avoiding the part k is 1/(1-z-...-z^(k-1) - z^(k+1)-...). - Gregory L. Simay, Sep 09 2018
If p,q,r are the three solutions to x^3 = 2x^2 - x + 1, then a(n) = (p-1)*p^n/((p-q)*(p-r)) + (q-1)*q^n/((q-p)*(q-r)) + (r-1)*r^n/((r-p)*(r-q)). - Greg Dresden and AnXing Yang, Aug 12 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.
C. 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, 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

More terms from Olivier Gérard, Jul 05 2001

A003622 The Wythoff compound sequence AA: a(n) = floor(n*phi^2) - 1, where phi = (1+sqrt(5))/2.

Original entry on oeis.org

1, 4, 6, 9, 12, 14, 17, 19, 22, 25, 27, 30, 33, 35, 38, 40, 43, 46, 48, 51, 53, 56, 59, 61, 64, 67, 69, 72, 74, 77, 80, 82, 85, 88, 90, 93, 95, 98, 101, 103, 106, 108, 111, 114, 116, 119, 122, 124, 127, 129, 132, 135, 137, 140, 142, 145, 148, 150, 153, 156, 158, 161, 163, 166

Offset: 1

N. J. A. Sloane, Mira Bernstein, Marc LeBrun

Also, integers with "odd" Zeckendorf expansions (end with ...+F_2 = ...+1) (Fibonacci-odd numbers); first column of Wythoff array A035513; from a 3-way splitting of positive integers. [Edited by Peter Munn, Sep 16 2022]
Also, numbers k such that A005206(k) = A005206(k+1). Also k such that A022342(A005206(k)) = k+1 (for all other k's this is k). - Michele Dondi (bik.mido(AT)tiscalenet.it), Dec 30 2001
Also, positions of 1's in A139764, the smallest term in Zeckendorf representation of n. - John W. Layman, Aug 25 2011
From Amiram Eldar, Sep 03 2022: (Start)
Numbers with an odd number of trailing 1's in their dual Zeckendorf representation (A104326), i.e., numbers k such that A356749(k) is odd.
The asymptotic density of this sequence is 1 - 1/phi (A132338). (End)
{a(n)} is the unique monotonic sequence of positive integers such that {a(n)} and {b(n)}: b(n) = a(n) - n form a partition of the nonnegative integers. - Yifan Xie, Jan 25 2025

A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 62.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 307-308 of 2nd edition.
C. Kimberling, "Stolarsky interspersions", Ars Combinatoria 39 (1995) 129-138.
D. R. Morrison, "A Stolarsky array of Wythoff pairs", in A Collection of Manuscripts Related to the Fibonacci Sequence. Fibonacci Assoc., Santa Clara, CA, 1980, pp. 134-136.
J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 10.
N. J. A. Sloane and Simon Plouffe, Encyclopedia of Integer Sequences, Academic Press, 1995: this sequence appears twice, as both M3277 and M3278.

A.H.M. Smeets, Table of n, a(n) for n = 1..20000 (terms 1.1000 from T. D. Noe)
J.-P. Allouche and F. M. Dekking, Generalized Beatty sequences and complementary triples, arXiv:1809.03424 [math.NT], 2018.
Jon Asier Bárcena-Petisco, Luis Martínez, María Merino, Juan Manuel Montoya, and Antonio Vera-López, Fibonacci-like partitions and their associated piecewise-defined permutations, arXiv:2503.19696 [math.CO], 2025. See p. 4.
A. Brousseau, Fibonacci and Related Number Theoretic Tables, Fibonacci Association, San Jose, CA, 1972, p. 62.
Larry Ericksen and Peter G. Anderson, Patterns in differences between rows in k-Zeckendorf arrays, The Fibonacci Quarterly, Vol. 50, February 2012. - _N. J. A. Sloane_, Jun 10 2012
Aviezri S. Fraenkel, The Raleigh game, INTEGERS: Electronic Journal of Combinatorial Number Theory 7.2 (2007): A13, 10 pages. See Table 1.
Martin Griffiths, On a Matrix Arising from a Family of Iterated Self-Compositions, Journal of Integer Sequences, 18 (2015), Article 15.11.8.
V. E. Hoggatt, Jr., 7-page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977.
Clark Kimberling, Interspersions.
Clark Kimberling, Complementary equations and Wythoff Sequences, JIS 11 (2008), Article 08.3.3.
Clark Kimberling, Lucas Representations of Positive Integers, J. Int. Seq., Vol. 23 (2020), Article 20.9.5.
Clark Kimberling, Intriguing infinite words composed of zeros and ones, Elemente der Mathematik (2021).
Clark Kimberling and K. B. Stolarsky, Slow Beatty sequences, devious convergence, and partitional divergence, Amer. Math. Monthly, 123 (No. 2, 2016), 267-273.
Johan Kok, Integer sequences with conjectured relation with certain graph parameters of the family of linear Jaco graphs, arXiv:2507.16500 [math.CO], 2025. See pp. 5-6.
L. Lindroos, A. Sills, and H. Wang, Odd fibbinary numbers and the golden ratio, Fib. Q., 52 (2014), 61-65.
A. J. Macfarlane, On the fibbinary numbers and the Wythoff array, arXiv:2405.18128 [math.CO], 2024. See page 3.
Mathematics Stack Exchange, Golden ratio and floor function floor(phi^2*n) - floor(phi*floor(phi*n)) = 1.
M. Rigo, P. Salimov, and E. Vandomme, Some Properties of Abelian Return Words, Journal of Integer Sequences, Vol. 16 (2013), Article 13.2.5.
N. J. A. Sloane, Classic Sequences
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)
Jiemeng Zhang, Zhixiong Wen, and Wen Wu, Some Properties of the Fibonacci Sequence on an Infinite Alphabet, Electronic Journal of Combinatorics, 24(2) (2017), Article P2.52.

Positions of 1's in A003849.
Complement of A022342.
Cf. A066096, A139764, A035516, A026273, A104326, A132338, A356749.
The Wythoff compound sequences: Let A = A000201, B = A001950. Then AA = A003622, AB = A003623, BA = A035336, BB = A101864. The eight triples AAA, AAB, ..., BBB are A134859, A134860, A035337, A134862, A134861, A134863, A035338, A134864, resp.
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A000201 as the parent: A000201, A001030, A001468, A001950, A003622, A003842, A003849, A004641, A005614, A014675, A022342, A088462, A096270, A114986, A124841. - N. J. A. Sloane, Mar 11 2021

a003622 n = a003622_list !! (n-1)
a003622_list = filter ((elem 1) . a035516_row) [1..]
-- Reinhard Zumkeller, Mar 10 2013

A003622 := proc(n)
    n+floor(n*(1+sqrt(5))/2)-1 ;
end proc: # R. J. Mathar, Jan 25 2015
# Maple code for the Wythoff compound sequences, from N. J. A. Sloane, Mar 30 2016
# The Wythoff compound sequences: Let A = A000201, B = A001950. Then AA = A003622, AB = A003623, BA = A035336, BB = A101864. The eight triples AAA, AAB, ..., BBB are A134859, A134860, A035337, A134862, A134861, A134863, A035338, A134864, resp.
# Assume files out1, out2 contain lists of the terms in the base sequences A and B from their b-files
read out1; read out2; b[0]:=b1: b[1]:=b2:
w2:=(i,j,n)->b[i][b[j][n]];
w3:=(i,j,k,n)->b[i][b[j][b[k][n]]];
for i from 0 to 1 do
lprint("name=",i);
lprint([seq(b[i][n],n=1..100)]):
od:
for i from 0 to 1 do for j from 0 to 1 do
lprint("name=",i,j);
lprint([seq(w2(i,j,n),n=1..100)]);
od: od:
for i from 0 to 1 do for j from 0 to 1 do for k from 0 to 1 do
lprint("name=",i,j,k);
lprint([seq(w3(i,j,k,n),n=1..100)]);
od: od: od:

With[{c=GoldenRatio^2},Table[Floor[n c]-1,{n,70}]] (* Harvey P. Dale, Jun 11 2011 *)
Range[70]//Floor[#*GoldenRatio^2]-1& (* Waldemar Puszkarz, Oct 10 2017 *)

PARI
```
a(n)=floor(n*(sqrt(5)+3)/2)-1
```

a(n) = (sqrtint(n^2*5)+n*3)\2 - 1; \\ Michel Marcus, Sep 17 2022

from sympy import floor
from mpmath import phi
def a(n): return floor(n*phi**2) - 1 # Indranil Ghosh, Jun 09 2017

from math import isqrt
def A003622(n): return (n+isqrt(5*n**2)>>1)+n-1 # Chai Wah Wu, Aug 11 2022

a(n) = floor(n*phi) + n - 1. [Corrected by Jianing Song, Aug 18 2022]
a(n) = floor(floor(n*phi)*phi) = A000201(A000201(n)). [See the Mathematics Stack Exchange link for a proof of the equivalence of the definition. - Jianing Song, Aug 18 2022]
a(n) = 1 + A022342(1 + A022342(n)).
G.f.: 1 - (1-x)*Sum_{n>=1} x^a(n) = 1/1 + x/1 + x^2/1 + x^3/1 + x^5/1 + x^8/1 + ... + x^F(n)/1 + ... (continued fraction where F(n)=n-th Fibonacci number). - Paul D. Hanna, Aug 16 2002
a(n) = A001950(n) - 1. - Philippe Deléham, Apr 30 2004
a(n) = A022342(n) + n. - Philippe Deléham, May 03 2004
a(n) = a(n-1) + 2 + A005614(n-2); also a(n) =  a(n-1) + 1 + A001468(n-1). - A.H.M. Smeets, Apr 26 2024

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

A002057 Fourth convolution of Catalan numbers: a(n) = 4binomial(2n+3,n)/(n+4).

Original entry on oeis.org

1, 4, 14, 48, 165, 572, 2002, 7072, 25194, 90440, 326876, 1188640, 4345965, 15967980, 58929450, 218349120, 811985790, 3029594040, 11338026180, 42550029600, 160094486370, 603784920024, 2282138106804, 8643460269248, 32798844771700, 124680849918352

Offset: 0

N. J. A. Sloane

a(n) is sum of the (flattened) list obtained by the iteration of: replace each integer k with the list 0,...,k+1 on the starting value 0. Length of this list is Catalan(n) or A000108. - Wouter Meeussen, Nov 11 2001
a(n-2) is the number of n-th generation vertices in the tree of sequences with unit increase labeled by 3 (cf. Zoran Sunic reference). - Benoit Cloitre, Oct 07 2003
Number of standard tableaux of shape (n+2,n-1). - Emeric Deutsch, May 30 2004
a(n) = CatalanNumber(n+3) - 2*CatalanNumber(n+2). Proof. From its definition as a convolution of Catalan numbers, a(n) counts lists of 4 Dyck paths of total size (semilength) = n. Connect the 4 paths by 3 upsteps (U) and append 3 downsteps (D). This is a reversible procedure. So a(n) is also the number of Dyck (n+3)-paths that end DDD (D for downstep). Let C(n) denote CatalanNumber(n) (A000108). Since C(n+3) is the total number of Dyck (n+3)-paths and C(n+2) is the number that end UD, we have (*) C(n+3) - C(n+2) is the number of Dyck (n+3)-paths that end DD. Also, (**) C(n+2) is the number of Dyck (n+3)-paths that end UDD (change the last D in a Dyck (n+2)-path to UDD). Subtracting (**) from (*) yields a(n) = C(n+3) - 2C(n+2) as claimed. - David Callan, Nov 21 2006
Convolution square of the Catalan sequence without one of the initial "1"'s: (1 + 4x + 14x^2 + 48x^3 + ...) = (1/x^2) * square(x + 2x^2 + 5x^3 + 14x^4 + ...)
a(n) is the number of binary trees with n+3 internal nodes in which both subtrees of the root are nonempty.  Cf. A068875 [Sedgewick and Flajolet]. - Geoffrey Critzer, Jan 05 2013
With offset 4, a(n) is the number of permutations on {1,2,...,n} that are 123-avoiding, i.e., do not contain a three-term monotone subsequence, for which the first ascent is at positions (4,5); for example, there are 48 123-avoiding permutations on n=7 for which the first ascent is at spots (4,5). See Connolly link. There it is shown in general that the k-th Catalan Convolution is the number of 123-avoiding permutations for which the first ascent is at (k, k+1). (For n=k, the first ascent is defined to be at positions (k,k+1) if the permutation is the decreasing permutation with no ascents.) - Anant Godbole, Jan 17 2014
With offset 4, a(n) is the number of permutations on {1,2,...,n} that are 123-avoiding and for which the integer n is in the 4th spot; see Connolly link. - Anant Godbole, Jan 17 2014
a(n) is the number of North-East lattice paths from (0,0) to (n+2,n+2) that have exactly one east step below the subdiagonal y = x-1. Details can be found in Section 3.1 in Pan and Remmel's link. - Ran Pan, Feb 04 2016
a(n) is the number of North-East lattice paths from (0,0) to (n+2,n+2) that bounce off the diagonal y = x to the right exactly once but do not bounce off y = x to the left. Details can be found in Section 4.2 in Pan and Remmel's link. - Ran Pan, Feb 04 2016
a(n) is the number of North-East lattice paths from (0,0) to (n+2,n+2) that horizontally cross the diagonal y = x exactly once but do not cross the diagonal vertically. Details can be found in Section 4.3 in Pan and Remmel's link. - Ran Pan, Feb 04 2016
Apparently also Young tableaux of (non-partition) shape [n+1, 1, 1, n+1], see example file. - Joerg Arndt, Dec 30 2023

			From _Peter Bala_, Apr 14 2017: (Start)
This sequence appears on the main diagonal of a generalized Catalan triangle. Construct a lower triangular array (T(n,k)), n,k >= 0 by placing the sequence [0,0,0,1,1,1,1,...] in the first column and then filling in the remaining entries in the array using the rule T(n,k) = T(n,k-1) + T(n-1,k). The resulting array begins
  n\k| 0 1  2  3  4   5   6   7  ...
  ---+-------------------------------
   0 | 0
   1 | 0 0
   2 | 0 0  0
   3 | 1 1  1  1
   4 | 1 2  3  4  4
   5 | 1 3  6 10 14  14
   6 | 1 4 10 20 34  48  48
   7 | 1 5 15 35 69 117 165 165
   ...
(see Tedford 2011; this is essentially the array C_4(n,k) in the notation of Lee and Oh). Compare with A279004. (End)

Pierre de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 11, coefficients of P_4(z).
C. Krishnamachary and M. Bheemasena Rao, Determinants whose elements are Eulerian, prepared Bernoullian and other numbers, J. Indian Math. Soc., Vol. 14 (1922), pp. 55-62, 122-138 and 143-146.
Robert Sedgewick and Phillipe Flajolet, Analysis of Algorithms, Addison Wesley, 1996, page 225.
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 and Fung Lam, Table of n, a(n) for n = 0..1000 (first 100 terms were computed by T. D. Noe)
Anwar Al Ghabra, K. Gopala Krishna, Patrick Labelle, and Vasilisa Shramchenko, Enumeration of multi-rooted plane trees, arXiv:2301.09765 [math.CO], 2023.
Joerg Arndt, The a(3)=48 Young tableaux for shape [4,1,1,4].
Andrei Asinowski and Cyril Banderier, From geometry to generating functions: rectangulations and permutations, arXiv:2401.05558 [cs.DM], 2024. See page 2.
Jean-Luc Baril and Helmut Prodinger, Enumeration of partial Lukasiewicz paths, arXiv:2205.01383 [math.CO], 2022.
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
Julia E. Bergner, Cedric Harper, Ryan Keller, and Mathilde Rosi-Marshall, Action graphs, planar rooted forests, and self-convolutions of the Catalan numbers, arXiv:1807.03005 [math.CO], 2018.
Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, Bounce statistics for rational lattice paths, arXiv:1707.09918 [math.CO], 2017, p. 9.
A. Cayley, On the partitions of a polygon, Proc. London Math. Soc., 22 (1891), 237-262 = Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 93ff.
Giulio Cerbai, Anders Claesson, and Luca Ferrari, Stack sorting with restricted stacks, arXiv:1907.08142 [cs.DS], 2019.
Gi-Sang Cheon, Hana Kim, and Louis W. Shapiro, Mutation effects in ordered trees, arXiv preprint arXiv:1410.1249 [math.CO], 2014 (see page 4).
Samuel Connolly, Zachary Gabor, and Anant Godbole, The location of the first ascent in a 123-avoiding permutation, arXiv:1401.2691 [math.CO], 2014.
S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin, and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., Vol. 35, No. 4 (1995) pp. 743-751.
S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin, and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., Vol. 35, No. 4 (1995) pp. 743-751. [Annotated scanned copy]
Olivier Danvy, Summa Summarum: Moessner's Theorem without Dynamic Programming, arXiv:2412.03127 [cs.DM], 2024. See p. 31.
Harold M. Edwards, A Normal Form for Elliptic Curves, Bulletin of the A.M.S., Vol. 44, No. 3 (2007), pp. 393-422.
Richard K. Guy, Letter to N. J. A. Sloane, May 1990
Richard K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seq., Vol. 3 (2000), Article 00.1.6.
V. E. Hoggatt, Jr., 7-page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977.
V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., Vol. 14, No. 5 (1976), pp. 395-405.
C. Krishnamachary and M. Bheemasena Rao, Determinants whose elements are Eulerian, prepared Bernoullian and other numbers, J. Indian Math. Soc., Vol. 14 (1922), pp. 55-62, 122-138 and 143-146. [Annotated scanned copy]
Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart., Vol. 38, No. 5 (2000), pp. 408-419. See Eq.(3).
Kyu-Hwan Lee and Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.
Nik Lygeros and Olivier Rozier, A new solution to the equation tau(rho) == 0 (mod p), J. Int. Seq., Vol. 13 (2010), Article 10.7.4.
Ran Pan and Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.
John Riordan, Letter to N. J. A. Sloane, Nov 10 1970.
John Riordan, Letter of 04/11/74.
L. W. Shapiro, A Catalan triangle, Discrete Math., Vol. 14, No. 1 (1976), pp. 83-90.
L. W. Shapiro, A Catalan triangle, Discrete Math., Vol. 14, No. 1 (1976), pp. 83-90. [Annotated scanned copy]
Zoran Sunic, Self describing sequences and the Catalan family tree, Elect. J. Combin., Vol. 10 (2003), Article N5.
Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016. See Appendix B2.
Steven J. Tedford, Combinatorial interpretations of convolutions of the Catalan numbers, Integers, Vol. 11 (2011), Article A3.
Wen-Jin Woan, Lou Shapiro, and D. G. Rogers, The Catalan numbers, the Lebesgue integral and 4^{n-2}, Amer. Math. Monthly, Vol. 104, No. 10 (1997), pp. 926-931.

T(n, n+4) for n=0, 1, 2, ..., array T as in A047072. Also a diagonal of A059365 and of A009766.
Cf. A001003.
A diagonal of any of the essentially equivalent arrays A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.
Cf. A000108, A000245, A000344, A003517, A000588, A003518, A003519, A001392, A001622.
Cf. A145596 (row sums), A279004.

List([0..25],n->4*Binomial(2*n+3,n)/(n+4)); # Muniru A Asiru, Mar 05 2018

[4*Binomial(2*n+3,n)/(n+4): n in [0..30]]; // Vincenzo Librandi, Feb 04 2016

a := n -> 32*4^n*GAMMA(5/2+n)*(1+n)/(sqrt(Pi)*GAMMA(5+n)):
seq(a(n),n=0..23); # Peter Luschny, Dec 14 2015
A002057List := proc(m) local A, P, n; A := [1]; P := [1,1,1];
for n from 1 to m - 2 do P := ListTools:-PartialSums([op(P), P[-1]]);
A := [op(A), P[-1]] od; A end: A002057List(27); # Peter Luschny, Mar 26 2022

Table[Plus@@Flatten[Nest[ #/.a_Integer:> Range[0, a+1]&, {0}, n]], {n, 0, 10}]
Table[4 Binomial[2n+3,n]/(n+4),{n,0,30}] (* or *) CoefficientList[ Series[ (1-Sqrt[1-4 x]+2 x (-2+Sqrt[1-4 x]+x))/(2 x^4),{x,0,30}],x] (* Harvey P. Dale, May 05 2011 *)

{a(n) = if( n<0, 0, n+=2; 2*binomial(2*n, n-2) / n)}; /* Michael Somos, Jul 31 2005 */

x='x+O('x^100); Vec((1-(1-4*x)^(1/2)+2*x*(-2+(1-4*x)^(1/2)+x))/(2*x^4)) \\ Altug Alkan, Dec 14 2015

[2*(n+1)*catalan_number(n+2)/(n+4) for n in (0..30)] # G. C. Greubel, May 27 2022

a(n) = A033184(n+4, 4) = 4*binomial(2*n+3, n)/(n+4) = 2*(n+1)*A000108(n+2)/(n+4).
G.f.: c(x)^4 with c(x) g.f. of A000108 (Catalan).
Row sums of A145596. Column 4 of A033184. By specializing the identities for the row polynomials given in A145596 we obtain the results a(n) = Sum_{k = 0..n} (-1)^k*binomial(n+1,k+1)*a(k)*4^(n-k) and a(n) = Sum_{k = 0..floor(n/2)} binomial(n+1,2*k+1) * Catalan(k+1) * 2^(n-2*k). From the latter identity we can derive the congruences a(2n+1) == 0 (mod 4) and a(2n) == Catalan(n+1) (mod 4). It follows that a(n) is odd if and only if n = (2^m - 4) for some m >= 2. - Peter Bala, Oct 14 2008
Let A be the Toeplitz matrix of order n defined by: A[i,i-1]=-1, A[i,j]=Catalan(j-i), (i<=j), and A[i,j]=0, otherwise. Then, for n>=3, a(n-3) = (-1)^(n-3) * coeff(charpoly(A,x), x^3). - Milan Janjic, Jul 08 2010
G.f.: (1-sqrt(1-4*x) + 2*x*(-2+sqrt(1-4*x) + x))/(2*x^4). - Harvey P. Dale, May 05 2011
a(n+1) = A214292(2*n+4,n). - Reinhard Zumkeller, Jul 12 2012
D-finite with recurrence: (n+4)a(n) = 8*(2*n-1)*a(n-3) - 20*(n+1)*a(n-2) + 4*(2*n+5)*a(n-1). - Fung Lam, Jan 29 2014
D-finite with recurrence: (n+4)*a(n) - 2*(3*n+7)*a(n-1) + 4*(2*n+1)*a(n-2) = 0. - R. J. Mathar, Jun 03 2014
Asymptotics: a(n) ~ 4^(n+3)/sqrt(4*Pi*n^3). - Fung Lam, Mar 31 2014
a(n) = 32*4^n*Gamma(5/2+n)*(1+n)/(sqrt(Pi)*Gamma(5+n)). - Peter Luschny, Dec 14 2015
a(n) = C(n+1) - 2*C(n) where C is Catalan number A000108. Yuchun Ji, Oct 18 2017 [Note: Offset is off by 2]
E.g.f.: d/dx ( 2*exp(2*x)*BesselI(2,2*x)/x ). - Ilya Gutkovskiy, Nov 01 2017
From Bradley Klee, Mar 05 2018: (Start)
With F(x) = 16/(1+sqrt(1-4*x))^4 g.f. of A002057, xi(x) = F(x/4)*(x/4)^2, K(16*x) = 2F1(1/2,1/2;1;16*x) g.f. of A002894, q(x) g.f. of A005797, and q'(x) g.f. of A274344:
  K(x) = (1+sqrt(xi(x)))*K(xi(x)).
  2*K(1-x) = (1+sqrt(xi(x)))*K(1-xi(x)).
  q(x) = sqrt(q(xi(16*x)/16)) = q'(xi(16*x)/16)/sqrt(xi(16*x)/16). (End)
From Amiram Eldar, Jan 02 2022: (Start)
Sum_{n>=0} 1/a(n) = 5/4 + Pi/(18*sqrt(3)).
Sum_{n>=0} (-1)^n/a(n) = 183*log(phi)/(25*sqrt(5)) - 77/100, where phi is the golden ratio (A001622). (End)
a(n) = Integral_{x=0..4} x^n*W(x) dx where W(x) = -x^(3/2)*(1 - x/2)*sqrt(4 - x)/Pi, defined on the open interval (0,4). - Karol A. Penson, Nov 13 2022

A003520 a(n) = a(n-1) + a(n-5); a(0) = ... = a(4) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 8, 11, 15, 20, 26, 34, 45, 60, 80, 106, 140, 185, 245, 325, 431, 571, 756, 1001, 1326, 1757, 2328, 3084, 4085, 5411, 7168, 9496, 12580, 16665, 22076, 29244, 38740, 51320, 67985, 90061, 119305, 158045, 209365, 277350, 367411, 486716, 644761

Offset: 0

N. J. A. Sloane

This comment covers a family of sequences which satisfy a recurrence of the form a(n) = a(n-1) + a(n-m), with a(n) = 1 for n = 0..m-1. The generating function is 1/(1-x-x^m). Also a(n) = Sum_{i=0..n/m} binomial(n-(m-1)*i, i). This family of binomial summations or recurrences gives the number of ways to cover (without overlapping) a linear lattice of n sites with molecules that are m sites wide. Special case: m=1: A000079; m=4: A003269; m=5: A003520; m=6: A005708; m=7: A005709; m=8: A005710.
Also counts ordered partitions such that no part is less than 5. For example, a(12) = a(11) + a(7) where a(7) counts 11,6+5 and 5+6 and a(11) counts 15,10+5, 9+6,8+7,7+8,6+9,5+10 and 5+5+5. Thus a(12) = 3 + 8 = 11. a(12) counts 16,11+5,10+6,9+7,8+8,7+9,6+10 and 6+5+5 but also 5+11,5+6+5 and 5+5+6. Similar results hold for the other sequences formed by a(n) = a(n-1) + a(n-k). - Alford Arnold, Aug 06 2003
Number of compositions of n into parts 1 and 5. - Joerg Arndt, Jun 25 2011
The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n>=5, 2*a(n-5) equals the number of 2-colored compositions of n with all parts >= 5, such that no adjacent parts have the same color. - Milan Janjic, Nov 27 2011
a(n+4) equals the number of binary words of length n having at least 4 zeros between every two successive ones. - Milan Janjic, Feb 07 2015
Number of tilings of a 5 X n rectangle with 5 X 1 pentominoes. - M. Poyraz Torcuk, Mar 26 2022

A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 119.
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..500
Jarib R. Acosta, Yadira Caicedo, Juan P. Poveda, José L. Ramírez, and Mark Shattuck, Some New Restricted n-Color Composition Functions, J. Int. Seq., Vol. 22 (2019), Article 19.6.4.
Mudit Aggarwal and Samrith Ram, Generating Functions for Straight Polyomino Tilings of Narrow Rectangles, J. Int. Seq., Vol. 26 (2023), Article 23.1.4.
Michael A. Allen, On a Two-Parameter Family of Generalizations of Pascal's Triangle, arXiv:2209.01377 [math.CO], 2022.
Michael A. Allen, Connections between Combinations Without Specified Separations and Strongly Restricted Permutations, Compositions, and Bit Strings, arXiv:2409.00624 [math.CO], 2024. See pp. 18, 22.
Roland Bacher, On the number of perfect lattices, arXiv:1704.02234 [math.NT], 2017. See Section 6.
D. Birmajer, J. B. Gil, and M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3, Example 9.
Bruce M. Boman, Geometric Capitulum Patterns Based on Fibonacci p-Proportions, Fibonacci Quart. 58 (2020), no. 5, 91-102.
Bruce M. Boman, Thien-Nam Dinh, Keith Decker, Brooks Emerick, Christopher Raymond, and Gilberto Schleinger, Why do Fibonacci numbers appear in patterns of growth in nature?, in Fibonacci Quarterly, 55(5): pp 30-41, (2017).
P. Chinn and S. Heubach, (1, k)-compositions, Congr. Numer. 164 (2003), 183-194. [Local copy]
E. Di Cera and Y. Kong, Theory of multivalent binding in one and two-dimensional lattices, Biophysical Chemistry, Vol. 61 (1996), pp. 107-124.
I. M. Gessel and Ji Li, Compositions and Fibonacci identities, J. Int. Seq. 16 (2013) 13.4.5
R. K. Guy, Letter to N. J. A. Sloane with attachment, 1988
V. C. Harris and C. C. Styles, A generalization of Fibonacci numbers, Fib. Quart. 2 (1964) 277-289, sequence u(n,4,1).
V. E. Hoggatt, Jr., 7-page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977.
Brian Hopkins and Hua Wang, Restricted Color n-color Compositions, arXiv:2003.05291 [math.CO], 2020.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 378
S. Kitaev, Independent sets on path-schemes, JIS 9 (2006) # 06.2.2 G(x) for M={1,2,3,4} gives seq. shifted 4 places left
T. G. Lewis, B. J. Smith and M. Z. Smith, Fibonacci sequences and money management, Fib. Quart., 14 (1976), 37-41.
Sergey Kirgizov, Q-bonacci words and numbers, arXiv:2201.00782 [math.CO], 2022.
R. J. Mathar, Paving rectangular regions with rectangular tiles: Tatami and Non-Tatami Tilings, arXiv:1311.6135 [math.CO], 2013, Table 33.
R. J. Mathar, Tiling n x m rectangles with 1 x 1 and s x s squares, arXiv:1609.03964 [math.CO] (2016), Section 4.4.
Augustine O. Munagi, Integer Compositions and Higher-Order Conjugation, J. Int. Seq., Vol. 21 (2018), Article 18.8.5.
Denis Neiter and Amsha Proag, Links Between Sums Over Paths in Bernoulli's Triangles and the Fibonacci Numbers, Journal of Integer Sequences, Vol. 19 (2016), Article 16.8.3.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
E. Wilson, The Scales of Mt. Meru
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1)

Apart from initial terms, same as A017899.

Cf. A000045, A000079, A000930, A003269, A005708, A005709, A005710, A005711.

a[0]:=1:a[1]:=1:a[2]:=1:a[3]:=1:a[4]:=1:for n from 5 to 60 do a[n]:=a[n-1]+a[n-5] od:seq(a[n],n=0..60);
with(combstruct): SeqSetU := [S, {S=Sequence(U), U=Set(Z, card > 4)}, unlabeled]: seq(count(SeqSetU, size=j), j=5..55); # Zerinvary Lajos, Oct 10 2006
A003520:=-1/(z**3+z**2-1)/(z**2-z+1); # Simon Plouffe in his 1992 dissertation
ZL:=[S, {a = Atom, b = Atom, S = Prod(X,Sequence(Prod(X,b))), X = Sequence(b,card >= 4)}, unlabelled]: seq(combstruct[count](ZL, size=n), n=4..54); # Zerinvary Lajos, Mar 26 2008
M := Matrix(5, (i,j)-> if j=1 then [1, 0, 0, 0, 1][i] elif (i=j-1) then 1 else 0 fi); a:= n-> (M^(n))[1,1]: seq(a(n), n=0..50); # Alois P. Heinz, Jul 27 2008

a[0] = a[1] = a[2] = a[3] = a[4] = 1; a[n_] := a[n] = a[n - 1] + a[n - 5]; Table[ a[n], {n, 0, 49}] (* Robert G. Wilson v, Dec 09 2004 *)
CoefficientList[Series[1/(1 - x - x^5), {x, 0, 51}], x] (* Zerinvary Lajos, Mar 29 2007 *)
LinearRecurrence[{1, 0, 0, 0, 1}, {1, 1, 1, 1, 1}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2012 *)
nxt[{a_,b_,c_,d_,e_}]:={b,c,d,e,e+a}; NestList[nxt,{1,1,1,1,1},50][[;;,1]] (* Harvey P. Dale, Sep 27 2023 *)

a(n):=sum(binomial(n-1+(-4)*j,j),j,0,(n-1)/4); /* Vladimir Kruchinin, May 23 2011 */

my(x='x+O('x^66)); Vec(x/(1-(x+x^5))) /* Joerg Arndt, Jun 25 2011 */

G.f.: 1/(1-x-x^5) = 1/((1-x+x^2)(1-x^2-x^3)).
a(n) = Sum_{j=0..(n-1)/4} binomial(n-1+(-4)*j,j).
For n>5, a(n) = floor( d*c^n + 1/2) where c is the positive real root of x^5-x^4-1 and d is the positive real root of 161*x^3-23*x^2-12*x-1 ( c=1.32471795724474602... and d=0.3811571478326847...) - Benoit Cloitre, Nov 30 2002
a(n) = term (1,1) in the 5 X 5 matrix [1,1,0,0,0; 0,0,1,0,0; 0,0,0,1,0; 0,0,0,0,1; 1,0,0,0,0]^n. - Alois P. Heinz, Jul 27 2008
For positive integers n and k such that k <= n <= 5*k, and 4 divides n-k, define c(n,k) = binomial(k,(n-k)/4), and c(n,k)=0, otherwise. Then, for n >= 1,  a(n) = sum(c(n,k), k=1..n). - Milan Janjic, Dec 09 2011
Apparently a(n) = hypergeometric([-1/5*n, 1/5-1/5*n, 2/5-1/5*n, 3/5-1/5*n, 4/5-1/5*n], [-1/4*n, 1/4-1/4*n, 1/2-1/4*n, 3/4-1/4*n], -5^5/4^4) for n>=16. - Peter Luschny, Sep 18 2014
7*a(n) = A117373(n+4) +5*b(n) +4*b(n-1) +b(n-2) where b(n) = A182097(n). - R. J. Mathar, Aug 07 2017

Additional comments from Yong Kong (ykong(AT)curagen.com), Dec 16 2000

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

A003517 Number of permutations of [n+1] with exactly 1 increasing subsequence of length 3.

Original entry on oeis.org

1, 6, 27, 110, 429, 1638, 6188, 23256, 87210, 326876, 1225785, 4601610, 17298645, 65132550, 245642760, 927983760, 3511574910, 13309856820, 50528160150, 192113383644, 731508653106, 2789279908316, 10649977831752, 40715807302800, 155851062397940, 597261490737912

Offset: 2

N. J. A. Sloane

a(n-4) = number of n-th generation vertices in the tree of sequences with unit increase labeled by 5 (cf. Zoran Sunic reference). - Benoit Cloitre, Oct 07 2003
Number of standard tableaux of shape (n+3,n-2). - Emeric Deutsch, May 30 2004
a(n) = A214292(2*n,n-3) for n > 2. - Reinhard Zumkeller, Jul 12 2012
a(n) is the number of North-East paths from (0,0) to (n,n) that cross the diagonal y = x horizontally exactly once. By symmetry, it is also the number of North-East paths from (0,0) to (n,n) that cross the diagonal y = x vertically exactly once. Details can be found in Section 3.3 in Pan and Remmel's link. - Ran Pan, Feb 02 2016
a(n) is the number of permutations pi of [n+3] such that s(pi)=321456...(n+3), where s denotes West's stack-sorting map. - Colin Defant, Jan 14 2019
a(n) is also the number of permutations of [n+1] avoiding the pattern 321. For permutations avoiding any of the other permutations of [3] (that is, any of 132, 213, 231, or 312) see A002054. - N. J. A. Sloane, Nov 26 2022

			a(3)=6 because the only permutations of 1234 with exactly 1 increasing subsequence of length 3 are 1423, 4123, 1342, 2314, 2341, 3124.

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 = 2..1000
Anwar Al Ghabra, K. Gopala Krishna, Patrick Labelle, and Vasilisa Shramchenko, Enumeration of multi-rooted plane trees, arXiv:2301.09765 [math.CO], 2023.
Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, Bounce statistics for rational lattice paths, arXiv:1707.09918 [math.CO], 2017, p. 9.
David Callan, A recursive bijective approach to counting permutations..., arXiv:math/0211380 [math.CO], 2002.
S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin, and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., Vol. 35, No. 4 (1995), pp. 743-751.
S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin, and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., Vol. 35, No. 4 (1995), pp. 743-751. [Annotated scanned copy]
Olivier Danvy, Summa Summarum: Moessner's Theorem without Dynamic Programming, arXiv:2412.03127 [cs.DM], 2024. See p. 31.
Hilmar Haukur Gudmundsson, Dyck paths, standard Young tableaux, and pattern avoiding permutations, PU. M. A., Vol. 21 , No. 2 (2010), pp. 265-284 (see 4.3 p. 277).
Richard K. Guy, Letter to N. J. A. Sloane, May 1990.
Richard K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article 00.1.6.
V. E. Hoggatt, Jr., 7-page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977.
V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., 14 (1976), 395-405.
Markus Kuba and Alois Panholzer, Stirling permutations containing a single pattern of length three, Australasian Journal of Combinatorics, Vol. 74, No. 2 (2019), pp. 216-239.
Nik Lygeros and Olivier Rozier, A new solution to the equation tau(rho) == 0 (mod p), J. Int. Seq., Vol. 13 (2010), Article 10.7.4.
John Noonan, The number of permutations containing exactly one increasing subsequence of length three, Discrete Math., Vol. 152, No. 1-3 (1996), pp. 307-313.
John Noonan and Doron Zeilberger, The Enumeration of Permutations With a Prescribed Number of "Forbidden" Patterns, arXiv:math/9808080 [math.CO], 1998.
Ran Pan and Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.
L. W. Shapiro, A Catalan triangle, Discrete Math., Vol. 14, No. 1 (1976), pp. 83-90.
L. W. Shapiro, A Catalan triangle, Discrete Math., Vol. 14, No. 1 (1976), pp. 83-90. [Annotated scanned copy]
Zoran Sunic, Self describing sequences and the Catalan family tree, Elect. J. Combin., Vol. 10 (2003), Article N5.

T(n, n+6) for n=0, 1, 2, ..., array T as in A047072.
See also A002054.
Cf. A001089, A084249, A000108, A000245, A002057, A000344, A000588, A003518, A003519, A001392, A001622, A214292.
First differences are in A026017.
A diagonal of any of the essentially equivalent arrays: A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.

A003517List := proc(m) local A, P, n; A := [1]; P := [1,1,1,1,1];
for n from 1 to m - 2 do P := ListTools:-PartialSums([op(P), P[-1]]);
A := [op(A), P[-1]] od; A end: A003517List(25); # Peter Luschny, Mar 26 2022

f[x_] = (Sqrt[1 - 4 x] - 1)^6/(64 x^4); CoefficientList[Series[f[x], {x, 0, 25}], x][[3 ;; 26]] (* Jean-François Alcover, Jul 13 2011, after g.f. *)
Table[6 Binomial[2n+1,n-2]/(n+4),{n,2,30}] (* Harvey P. Dale, Feb 27 2012 *)

a(n)=6*binomial(2*n+1,n-2)/(n+4) \\ Charles R Greathouse IV, May 18 2015

x='x+O('x^50); Vec(x^2*((1-(1-4*x)^(1/2))/(2*x))^6) \\ Altug Alkan, Nov 01 2015

a(n) = 6*binomial(2*n+1, n-2)/(n+4).
G.f.: x^2*C(x)^6, where C(x) is g.f. for the Catalan numbers (A000108). - Emeric Deutsch, May 30 2004
E.g.f.: exp(2*x)*(Bessel_I(2,2*x) - Bessel_I(4,2*x)). - Paul Barry, Jun 04 2007
Let A be the Toeplitz matrix of order n defined by: A[i,i-1]=-1, A[i,j]=Catalan(j-i), (i<=j), and A[i,j]=0, otherwise. Then, for n >= 5, a(n-3) = (-1)^(n-5)*coeff(charpoly(A,x),x^5). - Milan Janjic, Jul 08 2010
a(n) = Sum_{i>=1, j>=1, k>=1, i+j+k=n+1} Catalan(i)*Catalan(j)*Catalan(k). T. D. Noe, Dec 22 2010
D-finite with recurrence -(n+4)*(n-2)*a(n) + 2*n*(2*n+1)*a(n-1) = 0. - R. J. Mathar, Dec 04 2012
From Amiram Eldar, Jan 02 2022: (Start)
Sum_{n>=2} 1/a(n) = 7/2 - 34*Pi/(27*sqrt(3)).
Sum_{n>=2} (-1)^n/a(n) = 828*log(phi)/(25*sqrt(5)) - 2819/450, where phi is the golden ratio (A001622). (End)
a(n) ~ 3*4^(n+1)/(n^(3/2)*sqrt(Pi)). - Stefano Spezia, Apr 17 2024
a(n) = A000108(n+3) - 4*A000108(n+2) + 3*A000108(n+1). - Taras Goy, Jul 15 2024
a(n) = 6*(2*n+1)!*(n-1)!/((2*n-4)!*(n+4)!)*A000108(n-2). - Taras Goy, Dec 21 2024

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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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A002293 Number of dissections of a polygon: binomial(4*n, n)/(3*n + 1).

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A005251 a(0) = 0, a(1) = a(2) = a(3) = 1; thereafter, a(n) = a(n-1) + a(n-2) + a(n-4).

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

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

A002293 Number of dissections of a polygon: binomial(4n, n)/(3n + 1).

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

A002057 Fourth convolution of Catalan numbers: a(n) = 4binomial(2n+3,n)/(n+4).