A230390 -id:A230390 - cp's OEIS frontend

A234465 a(n) = 3binomial(8n+6,n)/(4*n+3).

1, 6, 63, 812, 11655, 178794, 2869685, 47593176, 809172936, 14028048650, 247039158366, 4406956913268, 79470057050020, 1446283758823470, 26529603944225670, 489989612605050800, 9104498753815680600, 170073237411754811568, 3192081704235788729043

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 = 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.
Wikipedia, Fuss-Catalan number

Cf. A000108, A007556, A234461, A234462, A234463, A234464, A234466, A234467, A230390, A007556, A069271, A118970, A212073, A233834, A234510, A234571, A235339.

[3*Binomial(8*n+6, n)/(4*n+3): n in [0..30]]; // Vincenzo Librandi, Dec 26 2013

Table[3 Binomial[8 n + 6, n]/(4 n + 3), {n, 0, 40}] (* Vincenzo Librandi, Dec 26 2013 *)

PARI
```
a(n) = 3*binomial(8*n+6,n)/(4*n+3);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(4/3))^6+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p = 8, r = 6.
O.g.f. A(x) = 1/x * series reversion (x/C(x)^6), where C(x) is the o.g.f. for the Catalan numbers A000108. A(x)^(1/6) is the o.g.f. for A007556. - Peter Bala, Oct 14 2015
D-finite with recurrence: 7*n*(7*n+4)*(7*n+1)*(7*n+5)*(7*n+2)*(7*n+6)*(7*n+3)*a(n) -128*(8*n+3)*(4*n-1)*(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

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)

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

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

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.

cp's OEIS Frontend

A234465 a(n) = 3*binomial(8*n+6,n)/(4*n+3).

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

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

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

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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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Formula

A234465 a(n) = 3binomial(8n+6,n)/(4*n+3).

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

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

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

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

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