A234505 a(n) = 2*binomial(9*n+2,n)/(9*n+2).

1, 2, 19, 252, 3885, 65274, 1159587, 21421248, 407337153, 7920326700, 156753610013, 3147328992080, 63951322669065, 1312575792628356, 27172514322677625, 566707337222428800, 11896007334177739113, 251142622845893276190, 5328891499524964282170

Offset: 0

Author

Tim Fulford, Dec 27 2013

Keywords

• nonn

Comments

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), where p=9, r=2.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906 [math.CO], 2007; Discrete Math., 308 (2008), 4660-4669.
• Thomas A. Dowling, Catalan Numbers Chapter 7
• Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.
• Jun Yan, Lattice paths enumerations weighted by ascent lengths, arXiv:2501.01152 [math.CO], 2025. See p. 7.
• Sheng-liang Yang and Mei-yang Jiang, Pattern avoiding problems on the hybrid d-trees, J. Lanzhou Univ. Tech., (China, 2023) Vol. 49, No. 2, 144-150. (in Mandarin)

Crossrefs

Cf. A000108, A143554, A234506, A234507, A234508, A234509, A234510, A234513, A232265.

Programs

Magma

[2*Binomial(9*n+2, n)/(9*n+2): n in [0..30]];

Mathematica

Table[2 Binomial[9 n + 2, n]/(9 n + 2), {n, 0, 30}]

PARI

a(n) = 2*binomial(9*n+2,n)/(9*n+2);

PARI

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(9/2))^2+x*O(x^n)); polcoeff(B, n)}

Formula

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=9, r=2.

a(n) = 2*binomial(9n+1,n-1)/n for n>0, a(0)=1. [Bruno Berselli, Jan 19 2014]