A196678 -id:A196678 - cp's OEIS frontend

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

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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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).

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P. Balduf, The propagator and diffeomorphisms of an interacting field theory, Master's thesis, submitted to the Institut für Physik, Mathematisch-Naturwissenschaftliche Fakultät, Humboldt-Universtät, Berlin, 2018.
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Paul Barry, d-orthogonal polynomials, Fuss-Catalan matrices and lattice paths, arXiv:2505.16718 [math.CO], 2025. See p. 2.
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D. Bevan, D. Levin, P. Nugent, J. Pantone, and L. Pudwell, Pattern avoidance in forests of binary shrubs, arXiv preprint arXiv:1510:08036 [math.CO], 2015.
Michel Bousquet and Cédric Lamathe, On symmetric structures of order two, Discrete Math. Theor. Comput. Sci. 10 (2008), 153-176.
T. Daniel Brennan, Christian Ferko, and Savdeep Sethi, A Non-Abelian Analogue of DBI from T₸, arXiv:1912.12389 [hep-th], 2019. See also SciPost Phys. Vol. 8 (2020), 052.
Wun-Seng Chou, Tian-Xiao He, and Peter J.-S. Shiue, On the Primality of the Generalized Fuss-Catalan Numbers, J. Int. Seqs., 21 (2018), #18.2.1.
Malin Christensson, Make hyperbolic tilings of images, web page, 2019.
J. Cigler, Some remarks about q-Chebyshev polynomials and q-Catalan numbers and related results, 2013.
N. Combe and V. Jugé, Counting bi-colored A'Campo forests, arXiv:1702.07672 [math.AG], 2017.
Tom Copeland, Discriminating Deltas, Depressed Equations, and Generalized Catalan Numbers, 2012.
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.
Isaac DeJager, Madeleine Naquin, and Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.
Bryan Ek, Lattice Walk Enumeration, arXiv:1803.10920 [math.CO], 2018.
Bryan Ek, Unimodal Polynomials and Lattice Walk Enumeration with Experimental Mathematics, arXiv:1804.05933 [math.CO], 2018.
Jishe Feng, The Hessenberg matrices and Catalan and its generalized numbers, arXiv:1810.09170 [math.CO], 2018. See p. 4.
M. Gross, Mirror symmetry and the Strominger-Yau-Zaslow conjecture, arXiv:1212.4220 [math.AG], p. 58, 2013.
F. Harary, E. M. Palmer, and R. C. Read, On the cell-growth problem for arbitrary polygons, computer printout, circa 1974.
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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.
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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

A234467 a(n) = 9binomial(8n + 9,n)/(8*n + 9).

Original entry on oeis.org

1, 9, 108, 1488, 22230, 350244, 5729724, 96395616, 1657248417, 28987537150, 514215324216, 9229030737264, 167283594343320, 3057857090083908, 56305821384711720, 1043424549990820800, 19445145508444588200, 364191559218548917713, 6851518654436447733980

Offset: 0

Tim Fulford, Dec 26 2013

Fuss-Catalan sequence is a(n,p,r) = r*binomial(n*p + r,n)/(n*p + r); this is the case p = 8, r = 9.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906 [math.CO], 2007.
J-C. Aval, Multivariate Fuss-Catalan Numbers, Discrete Math., 308 (2008), 4660-4669.
Thomas A. Dowling, Catalan Numbers Chapter 7
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.
Wikipedia, Fuss-Catalan number

Cf. A007556, A234461, A234462, A234463, A234464, A234465, A234466.

Cf. A000108, A000245 (k = 3), A006629 (k = 4), A196678 (k = 5), A233668 (k = 6), A233743 (k = 7), A233835 (k = 8), A232265 (k = 10), A229963 (k = 11).

[9*Binomial(8*n+9, n)/(8*n+9): n in [0..30]]; // Vincenzo Librandi, Dec 26 2013

Table[9 Binomial[8 n + 9, n]/(8 n + 9), {n, 0, 40}] (* Vincenzo Librandi, Dec 26 2013 *)

PARI
```
a(n) = 9*binomial(8*n+9,n)/(8*n+9);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(8/9))^9+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: A(x) = {1 + x*A(x)^(p/r)}^r, where p = 8, r = 9.
From Peter Bala, Oct 16 2015: (Start)
O.g.f.: (1/x) * series reversion (x*C(-x)^9), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108. See cross-references for other Fuss-Catalan sequences with o.g.f. 1/x * series reversion (x*C(-x)^k), k = 3 through 11.
A(x)^(1/9) is the o.g.f. for A007556. (End)
D-finite with recurrence +7*n*(7*n+3)*(7*n+4)*(7*n+5)*(7*n+6)*(7*n+8)*(7*n+9)*a(n)-128*(2*n+1)*(4*n+1)*(4*n+3)*(8*n+1)*(8*n+3)*(8*n+5)*(8*n+7)*a(n-1) = 0. - R. J. Mathar, Feb 09 2020
E.g.f.: F([9/8, 5/4, 11/8, 3/2, 13/8, 7/4, 15/8], [1, 10/7, 11/7, 12/7, 13/7, 15/7, 16/7], 16777216*x/823543), where F is the generalized hypergeometric function. - Stefano Spezia, Feb 09 2020

A229963 a(n) = 11binomial(10n + 11, n)/(10*n + 11) .

Original entry on oeis.org

1, 11, 165, 2860, 53900, 1072797, 22188859, 472214600, 10273141395, 227440759700, 5107663394691, 116068178638776, 2664012608972000, 61668340817988135, 1438101958237201950, 33753007927148177360, 796704536753910327114

Offset: 0

Tim Fulford, Oct 04 2013

Fuss-Catalan sequence is a(n,p,r) = r*binomial(n*p + r,n)/(n*p + r), where p = 10, r = 11.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906 [math.CO], 2007.
J-C. Aval, Multivariate Fuss-Catalan Numbers, Discrete Math., 308 (2008), 4660-4669.
Thomas A. Dowling, Catalan Numbers Chapter 7
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.
Wikipedia, Fuss-Catalan number

Cf. A000108, A059968, A234525, A234526, A234527, A234528, A234529, A234570, A234571, A234573.

Cf. A000245 (k = 3), A006629 (k = 4), A196678 (k = 5), A233668 (k = 6), A233743 (k = 7), A233835 (k = 8), A234467 (k = 9), A232265 (k = 10).

[11*Binomial(10*n+11,n)/(10*n+11) : n in [0..20]]; // Vincenzo Librandi, Jan 10 2014

Table[11/(10 n + 11) Binomial[10 n + 11, n], {n, 0, 40}] (* Vincenzo Librandi, Jan 10 2014 *)

a(n) = 11*binomial(10*n+11,n)/(10*n+11);

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(10/11))^11+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: A(x) = {1 + x*A(x)^(p/r)}^r, where p = 10, r = 11.
From _Peter Bala, Oct 16 2015: (Start)
O.g.f. A(x) = 1/x * series reversion (x*C(-x)^11), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108. See cross-references for other Fuss-Catalan sequences with o.g.f. 1/x * series reversion (x*C(-x)^k), k = 3 through 11.
A(x)^(1/11) is the o.g.f. for A059968. (End)
D-finite with recurrence: 81*n*(9*n+11)*(9*n+4)*(3*n+2)*(9*n+8)*(9*n+10)*(3*n+1)*(9*n+5)*(9*n+7)*a(n) -800*(10*n+1)*(5*n+1)*(10*n+3)*(5*n+2)*(2*n+1)*(5*n+3)*(10*n+7)*(5*n+4)*(10*n+9)*a(n-1)=0. - R. J. Mathar, Feb 21 2020

Corrected by Vincenzo Librandi, Jan 10 2014

A232265 a(n) = 10binomial(9n + 10, n)/(9*n + 10).

Original entry on oeis.org

1, 10, 135, 2100, 35475, 632502, 11714745, 223198440, 4346520750, 86128357150, 1731030945644, 35202562937100, 723029038312230, 14976976398326250, 312522428615310000, 6563314391270476752, 138617681440915119975, 2942332729799060033100, 62735156704285184848950

Offset: 0

Tim Fulford, Dec 28 2013

Fuss-Catalan sequence is a(n,p,r) = r*binomial(n*p + r,n)/(n*p + r), where p = 9, r = 10.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906 [math.CO], 2007.
J-C. Aval, Multivariate Fuss-Catalan Numbers, Discrete Math., 308 (2008), 4660-4669.
Thomas A. Dowling, Catalan Numbers Chapter 7
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.
Wikipedia, Fuss-Catalan number

Cf. A000108, A143554, A234505, A234506, A234507, A234508, A234509, A234510, A234513.

Cf. A062994, A000245 (k = 3), A006629 (k = 4), A196678 (k = 5), A233668 (k = 6), A233743 (k = 7), A233835 (k = 8), A234467 (k = 9), A229963 (k = 11).

[10*Binomial(9*n+10, n)/(9*n+10): n in [0..30]];

Table[10 Binomial[9 n + 10, n]/(9 n + 10), {n, 0, 30}]

PARI
```
a(n) = 10*binomial(9*n+10,n)/(9*n+10);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(9/10))^10+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: A(x) = {1 + x*A(x)^(p/r)}^r, where p = 9, r = 10.
From _Peter Bala, Oct 16 2015: (Start)
O.g.f. A(x) = 1/x * series reversion (x*C(-x)^10), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108. See cross-references for other Fuss-Catalan sequences with o.g.f. 1/x * series reversion (x*C(-x)^k), k = 3 through 11.
A(x)^(1/10) is the o.g.f. for A062994. (End)
D-finite with recurrence: 128*n*(8*n+3)*(4*n+3)*(8*n+9)*(2*n+1)*(8*n+7)*(4*n+5)*(8*n+5)*a(n) -81*(9*n+2)*(9*n+4)*(3*n+2)*(9*n+8)*(9*n+1)*(3*n+1)*(9*n+5)*(9*n+7)*a(n-1)=0. - R. J. Mathar, Feb 21 2020

A233835 a(n) = 8binomial(7n + 8, n)/(7*n + 8).

Original entry on oeis.org

1, 8, 84, 1008, 13090, 179088, 2542512, 37106784, 553270671, 8391423040, 129058047580, 2008018827360, 31550226597162, 499892684834368, 7978140653296800, 128138773298754240, 2069603881026760323, 33593111381834512200, 547698081896206040800, 8965330544164089648000, 147285313888568167177866

Offset: 0

Tim Fulford, Dec 16 2013

Fuss-Catalan sequence is a(n,p,r) = r*binomial(n*p + r, n)/(n*p + r); this is the case p = 7, r = 8.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906 [math.CO], 2007.
J-C. Aval, Multivariate Fuss-Catalan Numbers, Discrete Math., 308 (2008), 4660-4669.
Thomas A. Dowling, Catalan Numbers Chapter 7
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.
Wikipedia, Fuss-Catalan number

Cf. A000108, A002296, A233832, A233833, A143547, A233834, A233835, A233907, A233908.

Cf. A000245 (k = 3), A006629 (k = 4), A196678 (k = 5), A233668 (k = 6), A233743 (k = 7), A234467 (k = 9), A232265 (k = 10), A229963 (k = 11).

[8*Binomial(7*n+8, n)/(7*n+8): n in [0..30]];

Table[8 Binomial[7 n + 8, n]/(7 n + 8), {n, 0, 30}]

PARI
```
a(n) = 8*binomial(7*n+8,n)/(7*n+8);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(7/8))^8+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: A(x) = {1 + x*A(x)^(p/r)}^r, where p = 7, r = 8.
From _Peter Bala, Oct 16 2015: (Start)
O.g.f. A(x) = 1/x * series reversion (x*C(-x)^8), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108. See cross-references for other Fuss-Catalan sequences with o.g.f. 1/x * series reversion (x*C(-x)^k), k = 3 through 11.
A(x)^(1/8) is the o.g.f. for A002296. (End)

A233668 a(n) = 6binomial(5n + 6,n)/(5*n + 6).

Original entry on oeis.org

1, 6, 45, 380, 3450, 32886, 324632, 3290040, 34034715, 357919100, 3815041230, 41124015036, 447534498320, 4910258796240, 54257308779600, 603260892430960, 6744185681876505, 75764901779438850, 854867886710698755, 9683529727259434200

Offset: 0

Tim Fulford, Dec 14 2013

Fuss-Catalan sequence is a(n,p,r) = r*binomial(n*p + r, n)/(n*p + r); this is the case p = 5, r = 6.

C. H. Pah, M. R. Wahiddin, Combinatorial Interpretation of Raney Numbers and Tree Enumerations, Open Journal of Discrete Mathematics, 2015, 5, 1-9; http://www.scirp.org/journal/ojdm; http://dx.doi.org/10.4236/ojdm.2015.51001

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906 [math.CO], 2007.
J-C. Aval, Multivariate Fuss-Catalan Numbers, Discrete Math., 308 (2008), 4660-4669.
Thomas A. Dowling, Catalan Numbers Chapter 7
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.
Wikipedia, Fuss-Catalan number

Cf. A000108, A002294, A118969, A143546, A118971, A233669, A233736, A233737, A233738.

Cf. A000245 (k = 3), A006629 (k = 4), A196678 (k = 5), A233743 (k = 7), A233835 (k = 8), A234467 (k = 9), A232265 (k = 10), A229963 (k = 11).

[6*Binomial(5*n+6,n)/(5*n+6): n in [0..30]];

Table[6 Binomial[5 n + 6, n]/(5 n + 6), {n, 0, 30}]

PARI
```
a(n) = 6*binomial(5*n+6,n)/(5*n+6);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(5/6))^6+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: A(x) = {1 + x*A(x)^(p/r)}^r, here p = 5, r = 6.
From _Peter Bala, Oct 16 2015: (Start)
O.g.f. A(x) = 1/x * series reversion (x*C(-x)^6), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108. See cross-references for other Fuss-Catalan sequences with o.g.f. 1/x * series reversion (x*C(-x)^k), k = 3 through 11.
A(x)^(1/6) is the o.g.f. for A002294. (End)
D-finite with recurrence 8*n*(4*n+5)*(2*n+3)*(4*n+3)*a(n) -5*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*a(n-1)=0. - R. J. Mathar, Nov 22 2024

A233743 a(n) = 7binomial(6n + 7, n)/(6*n + 7).

Original entry on oeis.org

1, 7, 63, 644, 7105, 82467, 992446, 12271512, 154962990, 1990038435, 25909892008, 341225775072, 4537563627415, 60842326873230, 821692714673340, 11167153485624304, 152610018401940330, 2095863415900961490, 28910564819681953485, 400379714692751795820

Offset: 0

Tim Fulford, Dec 15 2013

Fuss-Catalan sequence is a(n,p,r) = r*binomial(n*p + r, n)/(n*p + r); this is the case p = 6, r = 7.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906 [math.CO], 2007.
J-C. Aval, Multivariate Fuss-Catalan Numbers, Discrete Math., 308 (2008), 4660-4669.
Thomas A. Dowling, Catalan Numbers Chapter 7
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.
Wikipedia, Fuss-Catalan number

Cf. A000108, A002295, A212071, A212072, A212073, A130564, A233827, A233829, A233830.

Cf. A000245 (k = 3), A006629 (k = 4), A196678 (k = 5), A233668 (k = 6), A233835 (k = 8), A234467 (k = 9), A232265 (k = 10), A229963 (k = 11).

[7*Binomial(6*n+7, n)/(6*n+7): n in [0..30]]; // Vincenzo Librandi, Dec 16 2013

Table[7 Binomial[6 n + 7, n]/(6 n + 7), {n, 0, 40}] (* Vincenzo Librandi, Dec 16 2013 *)

PARI
```
a(n) = 7*binomial(6*n+7,n)/(6*n+7);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(6/7))^7+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: A(x) = {1 + x*A(x)^(p/r)}^r, where p = 6, r = 7.
From _Peter Bala, Oct 16 2015: (Start)
O.g.f. A(x) = 1/x * series reversion (x*C(-x)^7), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108. See cross-references for other Fuss-Catalan sequences with o.g.f. 1/x * series reversion (x*C(-x)^k), k = 3 through 11.
A(x)^(1/7) is the o.g.f. for A002295. (End)

More terms from Vincenzo Librandi, Dec 16 2013

A233658 7binomial(4n + 7, n)/(4*n + 7).

Original entry on oeis.org

1, 7, 49, 357, 2695, 20930, 166257, 1344904, 11042724, 91801255, 771201431, 6536904290, 55838330730, 480197194260, 4154140621425, 36126361733616, 315647802951628, 2769544822393356, 24392874398953060, 215582307059144025, 1911286446370861455, 16993580092566979770, 151491588134469616215

Offset: 0

Tim Fulford, Dec 14 2013

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=4, r=7.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906v1, Discrete Math., 308 (2008), 4660-4669.
Thomas A. Dowling, Catalan Numbers Chapter 7
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.

Cf. A000108, A002293, A069271, A006632, A196678, A006633, A233666, A006634, A233667.

[7*Binomial(4*n+7,n)/(4*n+7): n in [0..30]];

Table[7 Binomial[4 n + 7, n]/(4 n + 7), {n, 0, 30}]

PARI
```
a(n) = 7*binomial(4*n+7,n)/(4*n+7);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(4/7))^7+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=4, r=7.
D-finite with recurrence 3*(3*n+5)*(3*n+7)*(n+2)*a(n) -(n+1)*(661*n^2+1301*n+558)*a(n-1) +120*(4*n+1)*(2*n+1)*(4*n-1)*a(n-2)=0. - R. J. Mathar, Nov 22 2024
D-finite with recurrence 3*n*(3*n+5)*(3*n+7)*(n+2)*a(n) -8*(4*n+5)*(2*n+3)*(4*n+3)*(n+1)*a(n-1)=0. - R. J. Mathar, Nov 22 2024

A233667 a(n) = 5binomial(4n+10,n)/(2*n+5).

Original entry on oeis.org

1, 10, 85, 700, 5750, 47502, 395560, 3321120, 28102425, 239503550, 2054455634, 17726454200, 153757722300, 1340045361750, 11729338225200, 103068670351552, 908923976461140, 8041606944709800, 71359997110169625, 634978885837495500, 5664526697522326590

Offset: 0

Tim Fulford, Dec 14 2013

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=4, r=10.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906v1, Discrete Math., 308 (2008), 4660-4669.
Thomas A. Dowling, Catalan Numbers Chapter 7
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.

Cf. A000108, A002293, A069271, A006632, A196678, A006633, A233658, A233666, A006634, A006635.

[5*Binomial(4*n+10,n)/(2*n+5): n in [0..30]];

Table[5 Binomial[4 n + 10, n]/(2 n + 5), {n, 0, 30}]

PARI
```
a(n) = 5*binomial(4*n+10,n)/(2*n+5);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(2/5))^10+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=4, r=10.
From Ilya Gutkovskiy, Sep 14 2018: (Start)
E.g.f.: 4F4(5/2,11/4,3,13/4; 1,11/3,4,13/3; 256*x/27).
a(n) ~ 5*2^(8*n+39/2)/(sqrt(Pi)*3^(3*n+21/2)*n^(3/2)). (End)

A233666 a(n) = 2binomial(4n + 8, n)/(n + 2).

Original entry on oeis.org

1, 8, 60, 456, 3542, 28080, 226548, 1855040, 15380937, 128896456, 1090119316, 9292881360, 79769043900, 688915123680, 5981962494852, 52193342019456, 457367224685012, 4023551800087200, 35521420783728880, 314608026125871720, 2794654131668318430

Offset: 0

Tim Fulford, Dec 14 2013

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=4, r=8.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906v1, Discrete Math., 308 (2008), 4660-4669.
Thomas A. Dowling, Catalan Numbers Chapter 7
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.

Cf. A000108, A002293, A069271, A006632, A196678, A006633, A233658, A006634, A233667, A006635.

[2*Binomial(4*n+8,n)/(n+2): n in [0..30]]; // Vincenzo Librandi, Dec 14 2013

Table[2/(n + 2) Binomial[4 n + 8, n], {n, 0, 40}] (* Vincenzo Librandi, Dec 14 2013 *)

PARI
```
a(n) = 4*binomial(4*n+8,n)/(n+2);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(1/2))^8+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=4, r=8.
From Ilya Gutkovskiy, Sep 14 2018: (Start)
E.g.f.: 4F4(2,9/4,5/2,11/4; 1,3,10/3,11/3; 256*x/27).
a(n) ~ 2^(8*n+35/2)/(sqrt(Pi)*3^(3*n+17/2)*n^(3/2)). (End)
D-finite with recurrence 3*(3*n+7)*(n+2)*(3*n+8)*a(n) -2*(n+1)*(317*n^2+954*n+709)*a(n-1) +112*(4*n+1)*(2*n+1)*(4*n+3)*a(n-2)=0. - R. J. Mathar, Nov 22 2024

cp's OEIS Frontend

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

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A234467 a(n) = 9*binomial(8*n + 9,n)/(8*n + 9).

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A229963 a(n) = 11*binomial(10*n + 11, n)/(10*n + 11) .

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A232265 a(n) = 10*binomial(9*n + 10, n)/(9*n + 10).

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A233835 a(n) = 8*binomial(7*n + 8, n)/(7*n + 8).

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A233668 a(n) = 6*binomial(5*n + 6,n)/(5*n + 6).

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

A234467 a(n) = 9binomial(8n + 9,n)/(8*n + 9).

A229963 a(n) = 11binomial(10n + 11, n)/(10*n + 11) .

A232265 a(n) = 10binomial(9n + 10, n)/(9*n + 10).

A233835 a(n) = 8binomial(7n + 8, n)/(7*n + 8).

A233668 a(n) = 6binomial(5n + 6,n)/(5*n + 6).

A233743 a(n) = 7binomial(6n + 7, n)/(6*n + 7).

A233658 7binomial(4n + 7, n)/(4*n + 7).

A233667 a(n) = 5binomial(4n+10,n)/(2*n+5).

A233666 a(n) = 2binomial(4n + 8, n)/(n + 2).