A234467 -id:A234467 - cp's OEIS frontend

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

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

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

A234464 5binomial(8n+5, n)/(8*n+5).

Original entry on oeis.org

1, 5, 50, 630, 8925, 135751, 2165800, 35759900, 605902440, 10475490875, 184068392508, 3277575482090, 59012418601500, 1072549882307925, 19651558477204200, 362592313327737592, 6731396321743423000, 125645122201355505000, 2356570385677427920770

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=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, A007556, A234461, A234462, A234463, A234465, A234466, A234467, A230390.

[5*Binomial(8*n+5, n)/(8*n+5): n in [0..30]]; // Vincenzo Librandi, Dec 26 2012

Table[5 Binomial[8 n + 5, n]/(8 n + 5), {n, 0, 40}] (* Vincenzo Librandi, Dec 26 2013 *)

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

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

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=8, r=5.

A230390 5binomial(8n+10,n)/(4*n+5).

Original entry on oeis.org

1, 10, 125, 1760, 26650, 423752, 6978510, 117998400, 2036685765, 35738059500, 635627275767, 11433154297760, 207621482341000, 3801296492623560, 70092637731997100, 1300500163756675200, 24262157874835233000, 454847339247972377850, 8564398318045559667475

Offset: 0

Tim Fulford, Dec 28 2013

nonn

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=8, 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, A007556, A234461, A234462, A234463, A234464, A234465, A234466, A234467.

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

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

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

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(4/5))^10+x*O(x^n)); polcoeff(B, n)}

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

cp's OEIS Frontend

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

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

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A234464 5*binomial(8*n+5, n)/(8*n+5).

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Magma

Mathematica

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PARI

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A230390 5*binomial(8*n+10,n)/(4*n+5).

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Programs

Magma

Mathematica

PARI

PARI

Formula

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

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

A234464 5binomial(8n+5, n)/(8*n+5).

A230390 5binomial(8n+10,n)/(4*n+5).