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A234461
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a(n) = binomial(8*n+2,n)/(4*n+1).
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9
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1, 2, 17, 200, 2728, 40508, 635628, 10368072, 174047640, 2987139122, 52177566870, 924548764752, 16578073731752, 300252605231600, 5484727796499708, 100933398334075824, 1869468985400220600, 34823332479175275600, 651947852922093741585
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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 = 2.
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REFERENCES
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Gi-Sang Cheon, S.-T. Jin, L. W. Shapiro, A combinatorial equivalence relation for formal power series, Linear Algebra and its Applications, Available online 30 March 2015.
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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
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 = 2.
a(n) = 2*binomial(8n+1,n-1)/n for n>0, a(0)=1. [Bruno Berselli, Jan 19 2014]
A(x^3) = 1/x * series reversion (x/C(x^3)^2), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108. A(x)^(1/2) is the o.g.f. for A007556. - Peter Bala, Oct 14 2015
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MATHEMATICA
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Table[Binomial[8 n + 2, n]/(4 n + 1), {n, 0, 30}]
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PROG
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(PARI) a(n) = binomial(8*n+2, n)/(4*n+1);
(PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^4)^2+x*O(x^n)); polcoeff(B, n)}
(MAGMA) [Binomial(8*n+2, n)/(4*n+1): n in [0..30]];
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CROSSREFS
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Cf. A000108, A007556, A234462, A234463, A234464, A234465, A234466, A234467, A230390.
Sequence in context: A242428 A199751 A126752 * A277768 A333990 A004029
Adjacent sequences: A234458 A234459 A234460 * A234462 A234463 A234464
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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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