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A054727
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Number of forests of rooted trees with n nodes on a circle without crossing edges.
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4
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1, 2, 7, 33, 181, 1083, 6854, 45111, 305629, 2117283, 14929212, 106790500, 773035602, 5652275723, 41683912721, 309691336359, 2315772552485, 17415395593371, 131632335068744, 999423449413828
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OFFSET
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1,2
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REFERENCES
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P. Flajolet and M. Noy, Analytic Combinatorics of Noncrossing Configurations, Discrete Math. 204 (1999), 203-229.
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LINKS
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Table of n, a(n) for n=1..20.
C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps, FPSAC02, Melbourne, 2002.
F. Cazals, Combinatorics of Non-Crossing Configurations, Studies in Automatic Combinatorics, Volume II (1997).
Source
Philippe Flajolet, Enumeration of planar configurations in computational geometry
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 486, 502
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FORMULA
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add(binomial(n, j - 1)*binomial(3*n - 2*j - 1, n - j)/(2*n - j), j = 1 .. n)
G.f. A(x) satisfies 2*A(x)^2=x*(1-sqrt(1-4*A(x)))*(1-A(x)). [From Vladimir Kruchinin, Nov 25 2011]
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MAPLE
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ZZ:=[F, {F=Union(Epsilon, ZB), ZB=Prod(Z1, P), P=Sequence(B), B=Prod(P, Z1, P), Z1=Prod(Z, F)}, unlabeled]: seq(count(ZZ, size=n), n=1..20); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2007
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MATHEMATICA
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a[n_] := (3*n-3)!/((n-1)!*(2*n-1)!)*HypergeometricPFQ[{1-2*n, 1-n, -n}, {3/2 - 3*n/2, 2 - 3*n/2}, -1/4]; Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Sep 05 2012, after formula *)
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CROSSREFS
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Cf. A006013.
Sequence in context: A162257 A214954 A055724 * A086618 A224769 A172387
Adjacent sequences: A054724 A054725 A054726 * A054728 A054729 A054730
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KEYWORD
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nonn
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AUTHOR
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Philippe Flajolet, Apr 20 2000
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STATUS
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approved
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