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A234462
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a(n) = 3*binomial(8*n+3,n)/(8*n+3).
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9
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1, 3, 27, 325, 4488, 67158, 1059380, 17346582, 292046040, 5023824887, 87915626370, 1560176040519, 28011228029512, 507874087572600, 9286024289123268, 171026036066072924, 3169969149156895800, 59085490354010508600, 1106795192170066119435
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OFFSET
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0,2
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COMMENTS
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Fuss-Catalan sequence is a(n,p,r) = r*binomial(n*p + r, n)/(n*p + r), this is the case p = 8, r = 3.
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LINKS
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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
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FORMULA
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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
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MATHEMATICA
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Table[3 Binomial[8 n + 3, n]/(8 n + 3), {n, 0, 40}] (* Vincenzo Librandi, Dec 26 2013 *)
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PROG
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(PARI) a(n) = 3/(8*n+3)*binomial(8*n+3, n);
(PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(8/3))^3+x*O(x^n)); polcoeff(B, n)}
(Magma) [3*Binomial(8*n+3, n)/(8*n+3): n in [0..30]]; // Vincenzo Librandi, Dec 26 2013
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CROSSREFS
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Cf. A000108, A007556, A234461, A234463, A234464, A234465, A234466, A234467, A230390.
Sequence in context: A078532 A264684 A217363 * A153853 A067000 A354658
Adjacent sequences: A234459 A234460 A234461 * A234463 A234464 A234465
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KEYWORD
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nonn,easy
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AUTHOR
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Tim Fulford, Dec 26 2013
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STATUS
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approved
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