A234465 -id:A234465 - cp's OEIS frontend

A007556 Number of 8-ary trees with n vertices.

1, 1, 8, 92, 1240, 18278, 285384, 4638348, 77652024, 1329890705, 23190029720, 410333440536, 7349042994488, 132969010888280, 2426870706415800, 44627576949364104, 826044435409399800, 15378186970730687400, 287756293703544823872, 5409093674555090316300

Offset: 0

N. J. A. Sloane

Shifts left when convolved three times.
From Wolfdieter Lang, Sep 14 2007: (Start)
a(n), n >= 1, enumerates octic (8-ary) trees (rooted, ordered, incomplete) with n vertices (including the root).
Pfaff-Fuss-Catalan sequence C^{m}_n for m = 8. See the Graham et al. reference, p. 347. eq. 7.66. See also the Pólya-Szegő reference.
Also 8-Raney sequence. See the Graham et al. reference, p. 346-7.
(End)
This is instance k = 8 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) = 8 octic trees (vertex degree less than or equal to 8 and 8 possible branchings) with 2 vertices (one of them the root). Adding one more branch (one more vertex) to these 8 trees yields 8*8 + binomial(8, 2) = 92 = 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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Harvey P. Dale, Table of n, a(n) for n = 0..750
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv:math/0205301 [math.CO], 2002]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
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.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 290
Lajos Takács, Enumeration of rooted trees and forests, Math. Scientist 18 (1993), 1-10, esp. Eq. (5).

Seventh column of triangle A062993.
Cf. A007318, A234461, A234462, A234463, A234464, A234465, A234466, A234467.
Cf. A130564.

a007556 0 = 1
a007556 n = a007318' (8 * n) (n - 1) `div` n
-- Reinhard Zumkeller, Jul 30 2013

[Binomial(8*n, n)/(7*n+1): n in [0..20]]; // Vincenzo Librandi, Apr 02 2015

seq(binomial(8*n+1,n)/(8*n+1),n=0..30); # Robert FERREOL, Apr 01 2015
n:=30: G:=series(RootOf(g = 1+x*g^8, g),x=0,n+1): seq(coeff(G,x,k),k=0..n); # Robert FERREOL, Apr 01 2015

Table[Binomial[8n, n]/(7n + 1), {n, 0, 20}] (* Harvey P. Dale, Dec 24 2012 *)

vector(100, n, n--; binomial(8*n, n)/(7*n+1)) \\ Altug Alkan, Oct 14 2015

a(n) = binomial(8*n, n)/(7*n+1) = binomial(8*n+1, n)/(8*n+1) = A062993(n+6,6).
O.g.f.: A(x) = 1 + x*A(x)^8 = 1/(1-x*A(x)^7).
a(0) = 1; a(n) = Sum_{i1 + i2 + .. i8 = n - 1} a(i1)*a(i2)*...*a(i8) for n >= 1. - Robert FERREOL, Apr 01 2015
a(n) = binomial(8*n, n - 1)/n for 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, Mar 26 2015: (Start)
In Maple notation,
e.g.f.: hypergeom([1/8, 1/4, 3/8, 1/2, 5/8, 3/4, 7/8], [2/7, 3/7, 4/7, 5/7, 6/7, 1, 8/7],(2^24/7^7)*z);
o.g.f.: hypergeom([1/8, 1/4, 3/8, 1/2, 5/8, 3/4, 7/8], [2/7, 3/7, 4/7, 5/7, 6/7, 8/7],(2^24/7^7)*z);
a(n) are special values of Jacobi polynomials, in Maple notation:
  a(n) = JacobiP(n - 1, 7*n + 1, -n, 1)/n, n = 1, 2, ...
(End)
From Peter Bala, Oct 14 2015: (Start)
A(x)^2 is o.g.f. for A234461; A(x)^3 is o.g.f. for A234462;
A(x)^4 is o.g.f. for A234463; A(x)^5 is o.g.f. for A234464;
A(x)^6 is o.g.f. for A234465; A(x)^7 is o.g.f. for A234466;
A(x)^9 is o.g.f. for A234467. (End)
a(n) ~ 2^(24*n + 1)/(sqrt(Pi)*7^(7*n + 3/2)*n^(3/2)). - Ilya Gutkovskiy, Feb 07 2017
D-finite with recurrence: 7*n*(7*n-3)*(7*n+1)*(7*n-2)*(7*n-5)*(7*n-1)*(7*n-4)*a(n) -128*(8*n-5)*(4*n-1)*(8*n-7)*(2*n-1)*(8*n-1)*(4*n-3)*(8*n-3)*a(n-1)=0. - R. J. Mathar, Feb 20 2020
G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^15). - Seiichi Manyama, Jun 16 2025

A234466 a(n) = 7binomial(8n+7,n)/(8*n+7).

Original entry on oeis.org

1, 7, 77, 1015, 14763, 228459, 3689595, 61474519, 1048927880, 18236463245, 321899509386, 5753527081211, 103922382296180, 1893943017506925, 34783258504651434, 643111366544129175, 11960812088346090200, 223614812152492437432, 4200107505573406222425

Offset: 0

Tim Fulford, Dec 26 2013

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=8, 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.
Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.

Cf. A000108, A007556, A118971, A130564, A234461, A234462, A234463, A234464, A234465, A234467, A230390.

[7*Binomial(8*n+7, n)/(8*n+7): n in [0..30]]; // Vincenzo Librandi, Dec 26 2013

Table[7 Binomial[8 n + 7, n]/(8 n + 7), {n, 0, 40}] (* Vincenzo Librandi, Dec 26 2013 *)

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

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

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=8, r=7.
E.g.f.: hypergeom([7, 9, 10, 11, 12, 13, 14]/8, [8, 9, 10, 11, 12, 13, 14]/7, (8^8/7^7)*x). Cf.: Ilya Gutkovskiy in A118971. - Wolfdieter Lang, Feb 06 2020
D-finite with recurrence: +7*(7*n+4)*(7*n+1)*(7*n+5)*(7*n+2)*(7*n+6)*(7*n+3)*(n+1)*a(n) -128*(8*n+3)*(4*n+3)*(8*n+1)*(2*n+1)*(8*n-1)*(4*n+1)*(8*n+5)*a(n-1)=0. - R. J. Mathar, Feb 21 2020
From Wolfdieter Lang, Feb 15 2024: (Start)
a(n) = binomial(8*n + 6, n+1)/(7*n + 6). This is instance k = 7 of c(k, n+1) given in a comment in A130564.
The compositional inverse of y*(1 - y)^7 is x*G(x), where G is the o.g.f.. That is, G(x)*(1 - x*G(x))^7 = 1. This is equivalent to the formula of the first line above with B = G. Take  A =  B^(1/7) then A*(1 - x*B) = 1 or B*(1 - x*B)^7 = 1.
The o.g.f is G(x) = 8F7([7..14]/8, [8..14]/7; (8^8/7^7)*x) =  (7/(8*x))*(1 - 7F6([-1,1,2,3,4,5,6]/8, [1,2,3,4,5,6]/7; (8^8/7^7)*x)). See the e.g.f. above.(End)

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

A212073 G.f. satisfies: A(x) = (1 + x*A(x)^(3/2))^4.

Original entry on oeis.org

1, 4, 30, 280, 2925, 32736, 383838, 4654320, 57887550, 734405100, 9467075926, 123648163392, 1632743088275, 21761329287600, 292362576381900, 3955219615609056, 53834425161872586, 736687428853685400, 10129401435828605700, 139876690363085200200

Offset: 0

Paul D. Hanna, Apr 29 2012

Fuss-Catalan sequence is a(n,p,r) = r*binomial(p*n + r, n)/(p*n + r); this is the case p = 6, r = 4. The o.g.f. B(x) of the Fuss_catalan sequence a(n,p,r) satisfies B(x) = {1 + x*B(x)^(p/r)}^r. - Peter Bala, Oct 14 2015

			G.f.: A(x) = 1 + 4*x + 30*x^2 + 280*x^3 + 2925*x^4 + 32736*x^5 +...
Related expansions:
A(x)^(3/2) = 1 + 6*x + 51*x^2 + 506*x^3 + 5481*x^4 +...+ A002295(n+1)*x^n +...
A(x)^(1/4) = 1 + x + 6*x^2 + 51*x^3 + 506*x^4 +...+ A002295(n)*x^n +...

Wikipedia, Fuss-Catalan number

Cf. A002295, A212071, A212072, A130564, A069271, A118970, A233834, A234465, A234510, A234571, A235339.

m = 20; A[_] = 0;
Do[A[x_] = (1 + x*A[x]^(3/2))^4 + O[x]^m, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Oct 20 2019 *)

{a(n)=binomial(6*n+4,n) * 4/(6*n+4)}
for(n=0, 40, print1(a(n), ", "))

{a(n)=local(A=1+4*x); for(i=1, n, A=(1+x*A^(3/2))^4+x*O(x^n)); polcoeff(A, n)}

a(n) = 4*binomial(6*n+4,n)/(6*n+4).
G.f. A(x) = G(x)^4 where G(x) = 1 + x*G(x)^6 is the g.f. of A002295.
O.g.f. A(x) = 1/x * series reversion (x/C(x)^4), where C(x) is the o.g.f. for the Catalan numbers A000108. - Peter Bala, Oct 14 2015
D-finite with recurrence 5*n*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*a(n) -72*(6*n-1)*(3*n-1)*(2*n+1)*(3*n+1)*(6*n+1)*a(n-1)=0. - R. J. Mathar, Nov 22 2024

A234461 a(n) = binomial(8n+2,n)/(4n+1).

Original entry on oeis.org

1, 2, 17, 200, 2728, 40508, 635628, 10368072, 174047640, 2987139122, 52177566870, 924548764752, 16578073731752, 300252605231600, 5484727796499708, 100933398334075824, 1869468985400220600, 34823332479175275600, 651947852922093741585

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 = 2.

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.
Gi-Sang Cheon, S.-T. Jin, and L. W. Shapiro, A combinatorial equivalence relation for formal power series, Linear Algebra and its Applications, Volume 491, 15 February 2016, Pages 123-137.
Thomas A. Dowling, Catalan Numbers Chapter 7
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.
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.
Wikipedia, Fuss-Catalan number

Cf. A000108, A007556, A234462, A234463, A234464, A234465, A234466, A234467, A230390.

[Binomial(8*n+2, n)/(4*n+1): n in [0..30]];

Table[Binomial[8 n + 2, n]/(4 n + 1), {n, 0, 30}]

PARI
```
a(n) = binomial(8*n+2,n)/(4*n+1);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^4)^2+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: A(x) = {1 + x*A(x)^(p/r)}^r, where p = 8, r = 2.
a(n) = 2*binomial(8n+1,n-1)/n for n>0, a(0)=1. [Bruno Berselli, Jan 19 2014]
A(x^3) = 1/x * series reversion (x/C(x^3)^2), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108. A(x)^(1/2) is the o.g.f. for A007556. - Peter Bala, Oct 14 2015

A234571 a(n) = 4binomial(10n+8,n)/(5*n+4).

Original entry on oeis.org

1, 8, 108, 1776, 32430, 632016, 12876864, 270964320, 5843355957, 128462407840, 2868356980060, 64869895026144, 1482877843096650, 34207542810153216, 795318309360948240, 18617396126132233920, 438423206616057162258, 10379232525028947311160, 246878659984195222962220

Offset: 0

Tim Fulford, Dec 28 2013

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

Cf. A059968, A234525, A234526, A234527, A234528, A234529, A234570, A234573, A059968, A069271, A118970, A212073, A233834, A234465, A234510, A235339.

[4*Binomial(10*n+8, n)/(5*n+4): n in [0..30]];

Table[4 Binomial[10 n + 8, n]/(5 n + 4), {n, 0, 30}]

PARI
```
a(n) = 4*binomial(10*n+8,n)/(5*n+4);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(5/4))^8+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p = 10, r = 8.

O.g.f. A(x) = 1/x * series reversion (x/C(x)^8), where C(x) is the o.g.f. for the Catalan numbers A000108. A(x)^(1/8) is the o.g.f. for A059968. - Peter Bala, Oct 14 2015

A234510 a(n) = 7binomial(9n+7,n)/(9*n+7).

Original entry on oeis.org

1, 7, 84, 1232, 20090, 349860, 6371764, 119877472, 2311664355, 45448324110, 907580289616, 18358110017520, 375353605696524, 7744997102466932, 161070300819384000, 3372697621463787456, 71046594621639707245, 1504569659175026591805

Offset: 0

Tim Fulford, Dec 27 2013

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

Cf. A000108, A143554, A234505, A234506, A234507, A234508, A234509, A234513, A232265, A062994, A069271, A118970, A212073, A233834, A234465, A234571, A235339.

[7*Binomial(9*n+7, n)/(9*n+7): n in [0..30]]; // Vincenzo Librandi, Dec 27 2013

Table[7 Binomial[9 n + 7, n]/(9 n + 7), {n, 0, 40}] (* Vincenzo Librandi, Dec 27 2013 *)

PARI
```
a(n) = 7*binomial(9*n+7,n)/(9*n+7);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(9/7))^7+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p = 9, r = 7.

O.g.f. A(x) = 1/x * series reversion (x/C(x)^7), where C(x) is the o.g.f. for the Catalan numbers A000108. A(x)^(1/7) is the o.g.f. for A062994. - Peter Bala, Oct 14 2015

A233834 a(n) = 5binomial(7n+5,n)/(7*n+5).

Original entry on oeis.org

1, 5, 45, 500, 6200, 82251, 1142295, 16398200, 241379325, 3623534200, 55262073757, 853814730600, 13335836817420, 210225027967325, 3340362288091500, 53443628421286320, 860246972339613855, 13921016318025200505, 226352372251889455000, 3696160728052814340000

Offset: 0

Tim Fulford, Dec 16 2013

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p = 7, r = 5.

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.
Wikipedia, Fuss-Catalan number

Cf. A000108, A002296, A233832, A233833, A143547, A130565, A233835, A233907, A233908, A002296, A069271, A118970, A212073, A234465, A234510, A234571, A235339.

[5*Binomial(7*n+5, n)/(7*n+5): n in [0..30]];

Table[5 Binomial[7 n + 5, n]/(7 n + 5), {n, 0, 30}]

PARI
```
a(n) = 5*binomial(7*n+5,n)/(7*n+5);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(7/5))^5+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: A(x) = {1 + x*A(x)^(p/r)}^r, where p = 7, r = 5.

O.g.f. A(x) = 1/x * series reversion (x/C(x)^5), where C(x) is the o.g.f. for the Catalan numbers A000108. A(x)^(1/5) is the o.g.f. for A002296. - Peter Bala, Oct 14 2015

A234463 Binomial(8n+4,n)/(2n+1).

Original entry on oeis.org

1, 4, 38, 468, 6545, 98728, 1566040, 25747128, 434824104, 7498246100, 131477423220, 2337053822012, 42016842044268, 762702138530080, 13959382918289880, 257323577200329904, 4773171937236245400, 89028543731246186400, 1668706597425638149302

Offset: 0

Tim Fulford, Dec 26 2013

nonn

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

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, A007556, A234461, A234462, A234464, A234465, A234466, A234467, A230390.

[Binomial(8*n+4, n)/(2*n+1): n in [0..30]]; // Vincenzo Librandi, Dec 26 2013

Table[Binomial[8 n + 4, n]/(2 n + 1), {n, 0, 40}] (* Vincenzo Librandi, Dec 26 2013 *)

PARI
```
a(n) = binomial(8*n+4,n)/(2*n+1);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^2)^4+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=8, r=4.

A234462 a(n) = 3binomial(8n+3,n)/(8*n+3).

Original entry on oeis.org

1, 3, 27, 325, 4488, 67158, 1059380, 17346582, 292046040, 5023824887, 87915626370, 1560176040519, 28011228029512, 507874087572600, 9286024289123268, 171026036066072924, 3169969149156895800, 59085490354010508600, 1106795192170066119435

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 = 3.

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
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.
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.
Wikipedia, Fuss-Catalan number

Cf. A000108, A007556, A234461, A234463, A234464, A234465, A234466, A234467, A230390.

[3*Binomial(8*n+3, n)/(8*n+3): n in [0..30]]; // Vincenzo Librandi, Dec 26 2013

Table[3 Binomial[8 n + 3, n]/(8 n + 3), {n, 0, 40}] (* Vincenzo Librandi, Dec 26 2013 *)

PARI
```
a(n) = 3/(8*n+3)*binomial(8*n+3,n);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(8/3))^3+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: A(x) = {1 + x*A(x)^(p/r)}^r, where p = 8, r = 3.

A(x^2) = 1/x * series reversion (x/C(x^2)^3), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108. A(x)^(1/3) is the o.g.f. for A007556. - Peter Bala, Oct 14 2015

cp's OEIS Frontend

A007556 Number of 8-ary trees with n vertices.

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A234466 a(n) = 7*binomial(8*n+7,n)/(8*n+7).

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

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A212073 G.f. satisfies: A(x) = (1 + x*A(x)^(3/2))^4.

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A234461 a(n) = binomial(8*n+2,n)/(4*n+1).

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A234571 a(n) = 4*binomial(10*n+8,n)/(5*n+4).

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A234466 a(n) = 7binomial(8n+7,n)/(8*n+7).

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

A234461 a(n) = binomial(8n+2,n)/(4n+1).

A234571 a(n) = 4binomial(10n+8,n)/(5*n+4).

A234510 a(n) = 7binomial(9n+7,n)/(9*n+7).

A233834 a(n) = 5binomial(7n+5,n)/(7*n+5).

A234463 Binomial(8n+4,n)/(2n+1).

A234462 a(n) = 3binomial(8n+3,n)/(8*n+3).