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A089436
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Number of non-crossing connected graphs on n nodes on a circle in which a fixed (distinguished) node has degree one.
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3
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1, 2, 9, 54, 374, 2820, 22485, 186494, 1592778, 13914108, 123750874, 1116809628, 10201516332, 94140605832, 876332565837, 8219124900558, 77594375595266, 736785675010380, 7031930543228910, 67420537625021460, 649070964647075700
(list; graph; refs; listen; history; internal format)
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OFFSET
| 2,2
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COMMENTS
| Convolution of (1, A007297) with itself.
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REFERENCES
| P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math. 204 (1999), 203-229.
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FORMULA
| G.f.=g^2, where g satisfies g^3+g^2-3zg+2z^2=0, g(0)=0, or, in Maple notation, g := -1/3+(2/3)*sqrt(1+9*z)*sin((1/3)*arcsin((2+27*z+54*z^2)/2/(1+9*z)^(3/2))).
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EXAMPLE
| a(3)=2 because among the four non-crossing graphs on the points A,B,C, the distinguished node A has degree equal to 1 only in the graphs {AB,BC} and {AC,BC}; in the other two graphs ({AB,AC} and {AB,BC,AC}) the node A has degree 2.
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MATHEMATICA
| max = 22; g[z_] := -3^(-1) + (2/3)*Sqrt[1 + 9*z]*Sin[(1/3)*ArcSin[(2 + 27*z + 54*z^2)/2/(1 + 9*z)^(3/2)]]; Drop[ CoefficientList[ Simplify[ Series[ g[z]^2, {z, 0, max}], z >= 0], z], 2] (* From Jean-François Alcover, Oct 05 2011 *)
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CROSSREFS
| Cf. A007297.
Sequence in context: A074602 A192131 A073986 * A000168 A127128 A064151
Adjacent sequences: A089433 A089434 A089435 * A089437 A089438 A089439
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KEYWORD
| nonn
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 28 2003
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