A234463 a(n) = binomial(8*n+4,n)/(2*n+1).

1, 4, 38, 468, 6545, 98728, 1566040, 25747128, 434824104, 7498246100, 131477423220, 2337053822012, 42016842044268, 762702138530080, 13959382918289880, 257323577200329904, 4773171937236245400, 89028543731246186400, 1668706597425638149302, 31414857910372264197100

Offset

0,2

Comments

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=8, r=4.

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=8, r=4.

a(n) ~ 2^(24*n+12) / (7^(7*n+9/2) * n^(3/2) * sqrt(Pi)). - Amiram Eldar, Sep 14 2025

Mathematica

Table[Binomial[8 n + 4, n]/(2 n + 1), {n, 0, 40}] (* Vincenzo Librandi, Dec 26 2013 *)

Prog

(PARI) a(n) = binomial(8*n+4, n)/(2*n+1);
(PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^2)^4+x*O(x^n)); polcoeff(B, n)}
(Magma) [Binomial(8*n+4, n)/(2*n+1): n in [0..30]]; // Vincenzo Librandi, Dec 26 2013

Crossrefs

Cf. A000108, A007556, A234461, A234462, A234464, A234465, A234466, A234467, A230390.

Sequence in context: A220748 A385719 A192947 * A194044 A317605 A263376

Adjacent sequences: A234460 A234461 A234462 * A234464 A234465 A234466

Keyword

nonn

Author

Tim Fulford, Dec 26 2013

Status

approved