A234525 a(n) = binomial(10*n+2,n)/(5*n+1).

1, 2, 21, 310, 5330, 99960, 1983049, 40919714, 869304150, 18885977110, 417663940540, 9371084905962, 212791660837756, 4880918206648000, 112925143575796455, 2632162372046272660, 61752662230350642670, 1457074607325333325524, 34554959743857577742575, 823201058861910748600350

Offset

0,2

Comments

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), where p=10, r=2.

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=10, r=2.

a(n) = 2*binomial(10n+1,n-1)/n for n > 0, a(0) = 1. - Bruno Berselli, Jan 19 2014

a(n) ~ 4^(5*n+1) * 5^(10*n+3/2) / (3^(18*n+5) * n^(3/2) * sqrt(Pi)). - Amiram Eldar, Sep 14 2025

Mathematica

Table[Binomial[10 n + 2, n]/(5 n + 1), {n, 0, 40}] (* Vincenzo Librandi, Dec 27 2013 *)

Prog

(PARI) a(n) = binomial(10*n+2, n)/(5*n+1);
(PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^5)^2+x*O(x^n)); polcoeff(B, n)}
(Magma) [Binomial(10*n+2, n)/(5*n+1): n in [0..30]]; // Vincenzo Librandi, Dec 27 2013

Crossrefs

Cf. A000108, A059968, A234526, A234527, A234528, A234529, A234570, A234571, A234573, A229963.

Sequence in context: A097627 A094797 A359918 * A230382 A189489 A378099

Adjacent sequences: A234522 A234523 A234524 * A234526 A234527 A234528

Keyword

nonn

Author

Tim Fulford, Dec 27 2013

Status

approved