A233833 a(n) = 3*binomial(7*n+3, n)/(7*n+3).

1, 3, 24, 253, 3045, 39627, 543004, 7718340, 112752783, 1682460520, 25533901536, 392912889915, 6116090678334, 96133810101609, 1523687678528400, 24324750346691480, 390786855500604195, 6313161418594235271, 102494297789621214400, 1671366110239940499000

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=3.

Links

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

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.

Formula

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=7, r=3.

From Ilya Gutkovskiy, Sep 14 2018: (Start)

E.g.f.: 6F6(3/7,4/7,5/7,6/7,8/7,9/7; 2/3,5/6,1,7/6,4/3,3/2; 823543*x/46656).

a(n) ~ 7^(7*n+5/2)/(sqrt(Pi)*3^(6*n+5/2)*4^(3*n+2)*n^(3/2)). (End)

Mathematica

Table[3 Binomial[7 n + 3, n]/(7 n + 3), {n, 0, 30}]

Prog

(PARI) a(n)=3*binomial(7*n+3, n)/(7*n+3);
(PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(7/3))^3+x*O(x^n)); polcoeff(B, n)}
(Magma) [3*Binomial(7*n+3, n)/(7*n+3): n in [0..30]];

Crossrefs

Cf. A000108, A002296, A233832, A143547, A233834, A130565, A233835, A233907, A233908.

Sequence in context: A203423 A319754 A218301 * A219536 A194957 A081133

Adjacent sequences: A233830 A233831 A233832 * A233834 A233835 A233836

Keyword

nonn

Author

Tim Fulford, Dec 16 2013

Status

approved