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A100300
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Number of dissections of a convex n-gon by nonintersecting diagonals into an odd number of regions.
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1
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1, 1, 6, 22, 99, 451, 2140, 10396, 51525, 259429, 1323362, 6824434, 35519687, 186346759, 984400760, 5231789176, 27954506505, 150079713481, 809181079294, 4379654830222, 23787413800491, 129607968854731, 708230837732436
(list; graph; refs; listen; history; internal format)
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OFFSET
| 3,3
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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
| a(n)=sum(binom(n-3, 2k-2)*binom(n+2k-3, 2k-2)/(2k-1), k=1..floor((n-1)/2)). G.f.=-(1/2)z^2/(1+z)+z/8+z^2/8-(z/8)sqrt(1-6*z+z^2).
(n-1)*(2*n-7)*a(n) = (2*n-5)*(5*n-19)*a(n-1)+(5*n-11)*(2*n-7)*a(n-2)-(2*n-5)*(n-5)*a(n-3). - Vladeta Jovovic (vladeta(AT)eunet.rs), Nov 12 2004
a(n) = (A001003(n-2)-(-1)^n)/2 = A100299(n)-(-1)^n, n>2. - Vladeta Jovovic (vladeta(AT)eunet.rs), Nov 15 2004
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EXAMPLE
| a(5)=6 because for a convex pentagon ABCDE we obtain dissections with an odd number of regions by one of the following sets of diagonals: {},{AC,AD}, {BD,BE}, {CE,CA}, {DA,DB} and {EB,EC}.
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MAPLE
| a:=n->sum(binomial(n-3, 2*k-2)*binomial(n+2*k-3, 2*k-2)/(2*k-1), k=1..floor((n-1)/2)): 1, seq(a(n), n=4..33);
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CROSSREFS
| Cf. A100299.
Sequence in context: A047124 A046365 A078418 * A027296 A179601 A151495
Adjacent sequences: A100297 A100298 A100299 * A100301 A100302 A100303
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KEYWORD
| nonn
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 12 2004
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