A233668 a(n) = 6*binomial(5*n + 6,n)/(5*n + 6).
1, 6, 45, 380, 3450, 32886, 324632, 3290040, 34034715, 357919100, 3815041230, 41124015036, 447534498320, 4910258796240, 54257308779600, 603260892430960, 6744185681876505, 75764901779438850, 854867886710698755, 9683529727259434200
Offset: 0
References
- C. H. Pah, M. R. Wahiddin, Combinatorial Interpretation of Raney Numbers and Tree Enumerations, Open Journal of Discrete Mathematics, 2015, 5, 1-9; http://www.scirp.org/journal/ojdm; http://dx.doi.org/10.4236/ojdm.2015.51001
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
- Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.
- Wikipedia, Fuss-Catalan number
Crossrefs
Programs
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Magma
[6*Binomial(5*n+6,n)/(5*n+6): n in [0..30]];
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Mathematica
Table[6 Binomial[5 n + 6, n]/(5 n + 6), {n, 0, 30}] -
PARI
a(n) = 6*binomial(5*n+6,n)/(5*n+6);
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PARI
{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(5/6))^6+x*O(x^n)); polcoeff(B, n)}
Formula
G.f. satisfies: A(x) = {1 + x*A(x)^(p/r)}^r, here p = 5, r = 6.
From _Peter Bala, Oct 16 2015: (Start)
O.g.f. A(x) = 1/x * series reversion (x*C(-x)^6), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108. See cross-references for other Fuss-Catalan sequences with o.g.f. 1/x * series reversion (x*C(-x)^k), k = 3 through 11.
A(x)^(1/6) is the o.g.f. for A002294. (End)
D-finite with recurrence 8*n*(4*n+5)*(2*n+3)*(4*n+3)*a(n) -5*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*a(n-1)=0. - R. J. Mathar, Nov 22 2024
Comments