A234465 -id:A234465 - cp's OEIS frontend

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

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.

A235339 a(n) = 9binomial(11n+9,n)/(11*n+9).

Original entry on oeis.org

1, 9, 135, 2460, 49725, 1072197, 24163146, 562311720, 13409091540, 325949656825, 8046743477058, 201198155083200, 5084704634041305, 129673310477725350, 3332952595603387800, 86250038091202771344, 2245329811618166111985

Offset: 0

Tim Fulford, Jan 06 2014

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p = 11, 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; 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. A230388, A234868, A234869, A234870, A234871, A234872, A234873, A235338, A235340, A069271, A118970, A212073, A230388, A233834, A234465, A234510, A234571.

[9*Binomial(11*n+9, n)/(11*n+9): n in [0..30]];

Table[9 Binomial[11 n + 9, n]/(11 n + 9), {n, 0, 30}]

PARI
```
a(n) = 9*binomial(11*n+9,n)/(11*n+9);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(11/9))^9+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, here p = 11, r = 9.

O.g.f. A(x) = 1/x * series reversion (x/C(x)^9), where C(x) is the o.g.f. for the Catalan numbers A000108. A(x)^(1/9) is the o.g.f. for A230388. - Peter Bala, Oct 14 2015

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.

A386396 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/7)} a(7k) a(n-1-7*k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 17, 27, 38, 50, 63, 77, 92, 200, 325, 468, 630, 812, 1015, 1240, 2728, 4488, 6545, 8925, 11655, 14763, 18278, 40508, 67158, 98728, 135751, 178794, 228459, 285384, 635628, 1059380, 1566040, 2165800, 2869685, 3689595, 4638348

Offset: 0

Seiichi Manyama, Jul 20 2025

nonn

Wikipedia, Fuss-Catalan number

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

Cf. A047749, A118968, A124753, A386379, A386380.

apr(n, p, r) = r*binomial(n*p+r, n)/(n*p+r);
a(n) = apr(n\7, 8, n%7+1);

For k=0..6, a(7*n+k) = (k+1) * binomial(8*n+k+1,n)/(8*n+k+1).

G.f. A(x) satisfies A(x) = 1/(1 - x * Product_{k=0..6} A(w^k*x)), where w = exp(2*Pi*i/7).

cp's OEIS Frontend

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

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A235339 a(n) = 9*binomial(11*n+9,n)/(11*n+9).

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

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Author

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Programs

Magma

Mathematica

PARI

PARI

Formula

A386396 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/7)} a(7*k) * a(n-1-7*k).

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Author

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Links

Crossrefs

Programs

PARI

Formula

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

A235339 a(n) = 9binomial(11n+9,n)/(11*n+9).

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

A386396 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/7)} a(7k) a(n-1-7*k).