A234466 -id:A234466 - 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.

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).

A386558 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = binomial((k+1)*n+k-1,n)/(n+1).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 7, 5, 0, 1, 4, 15, 30, 14, 0, 1, 5, 26, 91, 143, 42, 0, 1, 6, 40, 204, 612, 728, 132, 0, 1, 7, 57, 385, 1771, 4389, 3876, 429, 0, 1, 8, 77, 650, 4095, 16380, 32890, 21318, 1430, 0, 1, 9, 100, 1015, 8184, 46376, 158224, 254475, 120175, 4862, 0

Offset: 0

Seiichi Manyama, Jul 26 2025

			Square array begins:
  1,   1,    1,     1,      1,      1,       1, ...
  0,   1,    2,     3,      4,      5,       6, ...
  0,   2,    7,    15,     26,     40,      57, ...
  0,   5,   30,    91,    204,    385,     650, ...
  0,  14,  143,   612,   1771,   4095,    8184, ...
  0,  42,  728,  4389,  16380,  46376,  109668, ...
  0, 132, 3876, 32890, 158224, 548340, 1533939, ...

Wikipedia, Fuss-Catalan number

Columns k=0..10 give A000007, A000108, A006013, A006632, A118971, A130564(n+1), A130565(n+1), A234466, A234513, A234573, A235340.
Main diagonal gives A177784(n+1).
Cf. A162382.

a(n, k) = binomial((k+1)*n+k-1, n)/(n+1);

For k > 0, A(n,k) = r * binomial(n*p+r,n)/(n*p+r), the Fuss-Catalan number with p=k+1 and r=k.

G.f. of column k: (1/x) Series_Reverion( x*(1-x)^k ).

cp's OEIS Frontend

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

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

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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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A386558 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = binomial((k+1)*n+k-1,n)/(n+1).

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

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).