A233907 a(n) = 9*binomial(7*n+9, n)/(7*n+9).

1, 9, 99, 1218, 16065, 222138, 3178140, 46656324, 698868216, 10639125640, 164128169205, 2560224004884, 40314178429707, 639948824981928, 10230035192533800, 164541833894991240, 2660919275605834701, 43239781879996449825, 705687913212419321800, 11561996402992103418000, 190100812111989146008641

Offset

0,2

Comments

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=7, r=9.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Jean-Christophe Aval, Multivariate Fuss-Catalan Numbers, Discrete Math., Vol. 308, No. 20 (2008), 4660-4669; arXiv preprint, arXiv:0711.0906 [math.CO], 2007.

Thomas A. Dowling, Catalan Numbers, Chapter 7 of Applications of discrete mathematics, John G. Michaels and Kenneth H. Rosen (eds.), McGraw-Hill, New York, 1991. [Wayback Machine link]

Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Math. 15 (2010), 939-955.

Formula

G.f. satisfies: A(x) = (1 + x*A(x)^(p/r))^r, where p=7, r=9.

D-finite with recurrence 72*n*(6*n+5)*(3*n+2)*(2*n+3)*(3*n+4)*(6*n+7)*a(n) - 7*(7*n+4)*(7*n+8)*(7*n+5)*(7*n+2)*(7*n+6)*(7*n+3)*a(n-1) = 0. - R. J. Mathar, Nov 22 2024

a(n) ~ 7^(7*n+17/2) / (3^(6*n+15/2) * 4^(3*n+5) * n^(3/2) * sqrt(Pi)). - Amiram Eldar, Sep 15 2025

Mathematica

Table[9 Binomial[7 n + 9, n]/(7 n + 9), {n, 0, 30}]

Prog

(PARI) a(n) = 9*binomial(7*n+9, n)/(7*n+9);
(PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(7/9))^9+x*O(x^n)); polcoeff(B, n)}
(Magma) [9*Binomial(7*n+9, n)/(7*n+9): n in [0..30]];

Crossrefs

Cf. A000108, A002296, A233832, A233833, A143547, A233834, A130565, A233835, A233908.

Sequence in context: A264005 A232943 A245927 * A294120 A015685 A051581

Adjacent sequences: A233904 A233905 A233906 * A233908 A233909 A233910

Keyword

nonn

Author

Tim Fulford, Dec 17 2013

Status

approved