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A060941
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Duchon's numbers: the number of paths of length 5*n from the origin to the line y=2*x/3 with unit East and North steps that stay below the line or touch it.
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1
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1, 2, 23, 377, 7229, 151491, 3361598, 77635093, 1846620581, 44930294909, 1113015378438, 27976770344941, 711771461238122, 18293652115906958, 474274581883631615, 12388371266483017545, 325714829431573496525
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OFFSET
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0,2
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COMMENTS
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A generalization of the ballot numbers
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REFERENCES
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Cyril Banderier and Philippe Flajolet, Basic Analytic Combinatorics of Lattice Paths, Theoret. Comput. Sci. 281 (2002), 37-80.
Philippe Duchon, On the enumeration and generation of generalized Dyck words, Discrete Mathematics 225, 2000, 121-135
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LINKS
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Table of n, a(n) for n=0..16.
C. Banderier, Home page
Cyril Banderier, Philippe Flajolet, Basic Analytic Combinatorics of Lattice Paths, Theoret. Comput. Sci. 281 (2002), 37-80.
M. Bousqet-Melou and A. Jehanne, Polynomial equations with one catalytic variable, algebraic series and map enumeration
P. Duchon, Home Page
Philippe Duchon, On the enumeration and generation of generalized Dyck words, Discrete Mathematics 225, 2000, 121-135.
P. Flajolet, Home page
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FORMULA
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a(n) = sum(i=0..,n, 1/(5*n+i+1) * binomial(5*n+1, n-i) * binomial(5*n+2*i, i) ).
a(n) = sum(i=0..2*n, (-1)^i/(5*i+1) * binomial((5*i+1)/2, i) * 1/(1+5*(2*n-i)) * binomial((1+5*(2*n-i))/2, 2*n-i) ).
G.f. A(z) satisfies A(z) = 1+2*z^5*A^5-z^5*A^6+z^5*A^7+z^10*A^10.
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MATHEMATICA
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a[n_] = ((5n)!*(5n + 1)!*HypergeometricPFQRegularized[{-n, 5n/2 + 1/2, 5n/2 + 1}, {4n + 2, 5n + 2}, -4])/n!; a /@ Range[0, 16]
(* From Jean-François Alcover, Jun 30 2011, after given formula *)
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CROSSREFS
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Sequence in context: A091693 A211925 A197740 * A219890 A119774 A074649
Adjacent sequences: A060938 A060939 A060940 * A060942 A060943 A060944
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KEYWORD
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nice,nonn
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AUTHOR
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Philippe Flajolet, May 12 2001
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EXTENSIONS
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Updated Duchon URL - R. J. Mathar, Oct 01 2009
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STATUS
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approved
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