A234462 a(n) = 3*binomial(8*n+3,n)/(8*n+3).
1, 3, 27, 325, 4488, 67158, 1059380, 17346582, 292046040, 5023824887, 87915626370, 1560176040519, 28011228029512, 507874087572600, 9286024289123268, 171026036066072924, 3169969149156895800, 59085490354010508600, 1106795192170066119435
Offset: 0
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.
- J-C. Aval, Multivariate Fuss-Catalan Numbers, 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
Programs
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Magma
[3*Binomial(8*n+3, n)/(8*n+3): n in [0..30]]; // Vincenzo Librandi, Dec 26 2013
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Mathematica
Table[3 Binomial[8 n + 3, n]/(8 n + 3), {n, 0, 40}] (* Vincenzo Librandi, Dec 26 2013 *) -
PARI
a(n) = 3/(8*n+3)*binomial(8*n+3,n);
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PARI
{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(8/3))^3+x*O(x^n)); polcoeff(B, n)}
Formula
G.f. satisfies: A(x) = {1 + x*A(x)^(p/r)}^r, where p = 8, r = 3.
A(x^2) = 1/x * series reversion (x/C(x^2)^3), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108. A(x)^(1/3) is the o.g.f. for A007556. - Peter Bala, Oct 14 2015
Comments