A233668 a(n) = 6*binomial(5*n + 6,n)/(5*n + 6).

1, 6, 45, 380, 3450, 32886, 324632, 3290040, 34034715, 357919100, 3815041230, 41124015036, 447534498320, 4910258796240, 54257308779600, 603260892430960, 6744185681876505, 75764901779438850, 854867886710698755, 9683529727259434200

Offset

0,2

Comments

Fuss-Catalan sequence is a(n,p,r) = r*binomial(n*p + r, n)/(n*p + r); this is the case p = 5, r = 6.

References

C. H. Pah, M. R. Wahiddin, Combinatorial Interpretation of Raney Numbers and Tree Enumerations, Open Journal of Discrete Mathematics, 2015, 5, 1-9; http://www.scirp.org/journal/ojdm; http://dx.doi.org/10.4236/ojdm.2015.51001

Links

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

Formula

G.f. satisfies: A(x) = {1 + x*A(x)^(p/r)}^r, here p = 5, r = 6.

From _Peter Bala, Oct 16 2015: (Start)

O.g.f. A(x) = 1/x * series reversion (x*C(-x)^6), 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)^(1/6) is the o.g.f. for A002294. (End)

Mathematica

Table[6 Binomial[5 n + 6, n]/(5 n + 6), {n, 0, 30}]

Prog

(PARI) a(n) = 6*binomial(5*n+6, n)/(5*n+6);
(PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(5/6))^6+x*O(x^n)); polcoeff(B, n)}
(Magma) [6*Binomial(5*n+6, n)/(5*n+6): n in [0..30]];

Crossrefs

Cf. A000108, A002294, A118969, A143546, A118971, A233669, A233736, A233737, A233738.

Cf. A000245 (k = 3), A006629 (k = 4), A196678 (k = 5), A233743 (k = 7), A233835 (k = 8), A234467 (k = 9), A232265 (k = 10), A229963 (k = 11).

Sequence in context: A007194 A025551 A101600 * A243694 A227169 A365184

Adjacent sequences: A233665 A233666 A233667 * A233669 A233670 A233671

Keyword

nonn,easy

Author

Tim Fulford, Dec 14 2013

Status

approved