A234507 -id:A234507 - cp's OEIS frontend

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

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

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

A234513 8binomial(9n+8,n)/(9*n+8).

Original entry on oeis.org

1, 8, 100, 1496, 24682, 433160, 7932196, 149846840, 2898753715, 57135036024, 1143315429776, 23166186450680, 474347963242860, 9799792252101016, 204022381037886400, 4276098770070159096, 90151561242584838605, 1910564646571462338800

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=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.
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, A118971, A130564, A143554, A234466, A234505, A234506, A234507, A234508, A234509, A234510, A232265.

[8*Binomial(9*n+8, n)/(9*n+8): n in [0..30]]; // Vincenzo Librandi, Dec 28 2013

Table[8 Binomial[9 n + 8, n]/(9 n + 8), {n, 0, 30}] (* Vincenzo Librandi, Dec 28 2013 *)

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

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

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=9, r=8.
From Wolfdieter Lang, Feb 06 2020: (Start)
G.f.: hypergeom([8, 9, ..., 16]/9, [9, 10, ..., 16]/8, (9^9/8^8)*x).
E,g,f.: hypergeom([8, 10, 11, ..., 16]/9, [9, 10,..., 16]/8, (9^9/8^8)*x). Cf. Ilya Gutkovsky in  A118971. (End)
D-finite with recurrence 128*(8*n+3)*(4*n+3)*(8*n+1)*(2*n+1)*(8*n+7)*(4*n+1)*(8*n+5)*(n+1)*a(n) -81*(9*n+2)*(9*n+4)*(3*n+2)*(9*n-1)*(9*n+1)*(3*n+1)*(9*n+5)*(9*n+7)*a(n-1)=0. - R. J. Mathar, Aug 01 2022
From Wolfdieter Lang, Feb 15 2024: (Start)
a(n) = binomial(9*n+7, n+1)/(8*n+7), which is instance k = 8 of c(k, n+1) given in A130564.
The g.f. given above, and called B in the first line above, satisfies B(x)*(1 - x*B(x))^8 = 1. For the analog proof of the equivalence see A234466. x*B(x) is the compositional inverse of y*(1 - y)^8.
Another formula for the g.f. is B(x) =  (8/(9*x))*(1 - 8F7([-1,1,2,3,4,5,6.7]/9, [1,2,3,4,5,6.7]/8; (9^9/8^8)*x)). (End)

A234505 a(n) = 2binomial(9n+2,n)/(9*n+2).

Original entry on oeis.org

1, 2, 19, 252, 3885, 65274, 1159587, 21421248, 407337153, 7920326700, 156753610013, 3147328992080, 63951322669065, 1312575792628356, 27172514322677625, 566707337222428800, 11896007334177739113, 251142622845893276190, 5328891499524964282170

Offset: 0

Tim Fulford, Dec 27 2013

nonn

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), where p=9, 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; 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.
Jun Yan, Lattice paths enumerations weighted by ascent lengths, arXiv:2501.01152 [math.CO], 2025. See p. 7.
Sheng-liang Yang and Mei-yang Jiang, Pattern avoiding problems on the hybrid d-trees, J. Lanzhou Univ. Tech., (China, 2023) Vol. 49, No. 2, 144-150. (in Mandarin)

Cf. A000108, A143554, A234506, A234507, A234508, A234509, A234510, A234513, A232265.

[2*Binomial(9*n+2, n)/(9*n+2): n in [0..30]];

Table[2 Binomial[9 n + 2, n]/(9 n + 2), {n, 0, 30}]

PARI
```
a(n) = 2*binomial(9*n+2,n)/(9*n+2);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(9/2))^2+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=9, r=2.

a(n) = 2*binomial(9n+1,n-1)/n for n>0, a(0)=1. [Bruno Berselli, Jan 19 2014]

A234506 a(n) = binomial(9n+3, n)/(3n+1).

Original entry on oeis.org

1, 3, 30, 406, 6327, 107019, 1909908, 35399520, 674842149, 13147742322, 260626484118, 5239783981320, 106585537781775, 2189670831627678, 45366284782209600, 946815917066740800, 19887218367823853937, 420076689292591271325, 8917736795123409615060, 190161017612160607167948, 4071301730663135449185705

Offset: 0

Tim Fulford, Dec 27 2013

nonn

Fuss-Catalan sequence is a(n,p,r) = r*binomial(n*p + r, n)/(n*p + r), where p=9, r=3.

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, A143554, A232265, A234505, A234507, A234508, A234509, A234510, A234513.

[Binomial(9*n+3, n)/(3*n+1): n in [0..30]];

Table[Binomial[9n+3, n]/(3n+1), {n, 0, 30}]

PARI
```
a(n) = binomial(9*n+3,n)/(3*n+1);
```

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

[binomial(9*n+3, n)/(3*n+1) for n in (0..30)] # G. C. Greubel, Feb 09 2021

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=9, r=3.

A234508 5binomial(9n+5,n)/(9*n+5).

Original entry on oeis.org

1, 5, 55, 775, 12350, 211876, 3818430, 71282640, 1366368375, 26735839650, 531838637759, 10723307329700, 218658647805780, 4501362056183300, 93426735902060000, 1952884185072496992, 41074876852203972645, 868669222741822476975, 18460669540059117038250, 394033629095915025876750, 8443512680148379948569910

Offset: 0

Tim Fulford, Dec 27 2013

nonn

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

Cf. A000108, A143554, A234505, A234506, A234507, A234509, A234510, A234513, A232265.

[5*Binomial(9*n+5, n)/(9*n+5): n in [0..30]];

Table[5 Binomial[9 n + 5, n]/(9 n + 5), {n, 0, 30}]

PARI
```
a(n) = 5*binomial(9*n+5,n)/(9*n+5);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(9/5))^5+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=9, r=5.

A234509 2binomial(9n+6,n)/(3*n+2).

Original entry on oeis.org

1, 6, 69, 992, 15990, 276360, 5006386, 93817152, 1803606255, 35373572460, 704995403541, 14236901646240, 290687378847684, 5990903682047592, 124463414269524000, 2603845580096662656, 54807372993836345589, 1159856934027109448130, 24663454505518980363102, 526708243449729452311200, 11291926596343014148087470

Offset: 0

Tim Fulford, Dec 27 2013

nonn

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), where p=9, r=6.

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, A143554, A234505, A234506, A234507, A234508, A234510, A234513, A232265.

[2*Binomial(9*n+6, n)/(3*n+2): n in [0..30]];

Table[6 Binomial[9 n + 6, n]/(9 n + 6), {n, 0, 30}]

PARI
```
a(n) = 2*binomial(9*n+6,n)/(3*n+2);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(3/2))^6+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=9, r=6.

A386392 a(n) = 4 * binomial(7n+4,n)/(7n+4).

Original entry on oeis.org

1, 4, 34, 368, 4495, 59052, 814506, 11633440, 170574723, 2552698720, 38832808586, 598724403680, 9335085772194, 146936230074004, 2331703871687400, 37263447339612480, 599206511767593099, 9688121925389895636, 157401957319775436400, 2568427016865897264000

Offset: 0

Seiichi Manyama, Jul 20 2025

nonn

Wikipedia, Fuss-Catalan number

Cf. A118971, A212073, A234463, A234507, A234527, A234870.

Cf. A002296, A386380.

apr(n, p, r) = r*binomial(n*p+r, n)/(n*p+r);
a(n) = apr(n, 7, 4);

a(n) = r * binomial(n*p+r,n)/(n*p+r), the Fuss-Catalan number with p=7 and r=4.
a(n) = A386380(6*n+3).
G.f. A(x) satisfies A(x) = (1 + x*A(x)^(p/r))^r, where p=7, r=4.
G.f.: B(x)^4, where B(x) is the g.f. of A002296.

cp's OEIS Frontend

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

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

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A234513 8*binomial(9*n+8,n)/(9*n+8).

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A234505 a(n) = 2*binomial(9*n+2,n)/(9*n+2).

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A234506 a(n) = binomial(9*n+3, n)/(3*n+1).

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A234508 5*binomial(9*n+5,n)/(9*n+5).

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

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

A234513 8binomial(9n+8,n)/(9*n+8).

A234505 a(n) = 2binomial(9n+2,n)/(9*n+2).

A234506 a(n) = binomial(9n+3, n)/(3n+1).

A234508 5binomial(9n+5,n)/(9*n+5).

A234509 2binomial(9n+6,n)/(3*n+2).

A386392 a(n) = 4 * binomial(7n+4,n)/(7n+4).