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A196678
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a(n) = 5*binomial(4*n+5,n)/(4*n+5).
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13
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1, 5, 30, 200, 1425, 10626, 81900, 647280, 5217300, 42724825, 354465254, 2973052680, 25168220350, 214762810500, 1845308367000, 15951899986272, 138638564739180, 1210677947695620, 10617706139119000, 93477423115076000
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OFFSET
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0,2
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COMMENTS
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This is a sequence of power moments of the following signed function defined on the segment (0,256/27), in Maple notation:
-(1/2)*sqrt(2)*x^(1/4)*hypergeom([-5/12, -1/12, 5/4], [1/2, 3/4], (27/256)*x)/Pi+(5/4)*sqrt(x)*hypergeom([-1/6, 1/6, 3/2], [3/4, 5/4], (27/256)*x)/Pi-(15/64)*sqrt(2)*x^(3/4)*hypergeom([1/12, 5/12, 7/4], [5/4, 3/2], (27/256)*x)/Pi. This function is not positive on (0,256/27).
The two parameter Fuss-Catalan sequence is A(n,p,r) := r*binomial(n*p + r, n)/(n*p + r). This sequence is A(n,4,5). - Peter Bala, Oct 16 2015
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REFERENCES
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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
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LINKS
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FORMULA
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O.g.f.: hypergeom([5/4, 3/2, 7/4], [7/3, 8/3], (256 z)/27)
E.g.f.: hypergeom([5/4, 3/2, 7/4], [1, 7/3, 8/3], (256 z)/27)
From _Peter Bala, Oct 16 2015: (Start)
O.g.f. A(x) = 1/x * series reversion (x*C(-x)^5), 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)^(1/5) is the o.g.f. for A002293. (End)
D-finite with recurrence 3*n*(3*n+5)*(3*n+4)*a(n) -8*(4*n+1)*(2*n+1)*(4*n+3)*a(n-1)=0. - R. J. Mathar, Aug 01 2022
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PROG
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(Magma) [5*Binomial(4*n+5, n)/(4*n+5): n in [0..30]]; // Vincenzo Librandi, Oct 07 2011
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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