A233832 a(n) = 2*binomial(7*n+2,n)/(7*n+2).
1, 2, 15, 154, 1827, 23562, 320866, 4540200, 66096459, 983592304, 14894775896, 228784720710, 3555866673450, 55819631671902, 883738853546472, 14094715154157680, 226245021605612955, 3652242142988400570, 59254515909624764575, 965678197027521177200
Offset: 0
Keywords
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.
- Wikipedia, Fuss-Catalan number
- 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)
Programs
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Magma
[2*Binomial(7*n+2, n)/(7*n+2): n in [0..30]];
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Mathematica
Table[2 Binomial[7 n + 2, n]/(7 n + 2), {n, 0, 30}] -
PARI
a(n) = 2*binomial(7*n+2,n)/(7*n+2);
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PARI
{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(7/2))^2+x*O(x^n)); polcoeff(B, n)}
Formula
G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=7, r=2.
a(n) = 2*binomial(7n+1, n-1)/n for n>0, a(0)=1. [Bruno Berselli, Jan 19 2014]
From Ilya Gutkovskiy, Sep 14 2018: (Start)
E.g.f.: 6F6(2/7,3/7,4/7,5/7,6/7,8/7; 1/2,2/3,5/6,1,7/6,4/3; 823543*x/46656).
a(n) ~ 7^(7*n+3/2)/(sqrt(Pi)*3^(6*n+5/2)*4^(3*n+1)*n^(3/2)). (End)
Comments