A233658 a(n) = 7*binomial(4*n + 7, n)/(4*n + 7).

1, 7, 49, 357, 2695, 20930, 166257, 1344904, 11042724, 91801255, 771201431, 6536904290, 55838330730, 480197194260, 4154140621425, 36126361733616, 315647802951628, 2769544822393356, 24392874398953060, 215582307059144025, 1911286446370861455, 16993580092566979770, 151491588134469616215

Offset

0,2

Comments

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=4, r=7.

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=4, r=7.

From R. J. Mathar, Nov 22 2024: (Start)

D-finite with recurrence 3*(3*n+5)*(3*n+7)*(n+2)*a(n) - (n+1)*(661*n^2+1301*n+558)*a(n-1) +120*(4*n+1)*(2*n+1)*(4*n-1)*a(n-2) = 0.

D-finite with recurrence 3*n*(3*n+5)*(3*n+7)*(n+2)*a(n) - 8*(4*n+5)*(2*n+3)*(4*n+3)*(n+1)*a(n-1) = 0. (End)

a(n) ~ 7 * 2^(8*n+25/2) / (3^(3*n+15/2) * n^(3/2) * sqrt(Pi)). - Amiram Eldar, Sep 16 2025

Mathematica

Table[7 Binomial[4 n + 7, n]/(4 n + 7), {n, 0, 30}]

Prog

(PARI) a(n) = 7*binomial(4*n+7, n)/(4*n+7);
(PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(4/7))^7+x*O(x^n)); polcoeff(B, n)}
(Magma) [7*Binomial(4*n+7, n)/(4*n+7): n in [0..30]];

Crossrefs

Cf. A000108, A002293, A069271, A006632, A196678, A006633, A233666, A006634, A233667.

Sequence in context: A024587 A351057 A240721 * A344270 A144820 A344251

Adjacent sequences: A233655 A233656 A233657 * A233659 A233660 A233661

Keyword

nonn,easy

Author

Tim Fulford, Dec 14 2013

Status

approved