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A100300 Number of dissections of a convex n-gon by nonintersecting diagonals into an odd number of regions. 2
1, 1, 6, 22, 99, 451, 2140, 10396, 51525, 259429, 1323362, 6824434, 35519687, 186346759, 984400760, 5231789176, 27954506505, 150079713481, 809181079294, 4379654830222, 23787413800491, 129607968854731, 708230837732436, 3880366912218772, 21312485647242829, 117321536967959341 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,3

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 3..200

P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204, 203-229, 1999.

FORMULA

a(n) = sum(C(n-3, 2k-2)*C(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, Nov 12 2004

a(n) = (A001003(n-2)-(-1)^n)/2 = A100299(n)-(-1)^n, n>2. - Vladeta Jovovic, Nov 15 2004

Asymptotic (same as for A100299): a(n) ~ sqrt(3*sqrt(2)-4)*(3+2*sqrt(2))^(n-1)/(8*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 17 2012

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}.

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);

MATHEMATICA

Take[CoefficientList[Series[-1/2*x^2/(1+x)+x/8+x^2/8-x/8*Sqrt[1-6*x+x^2], {x, 0, 20}], x], {4, -1}] (* Vaclav Kotesovec, Oct 17 2012 *)

PROG

(PARI) x='x+O('x^66); Vec(-(1/2)*x^2/(1+x)+x/8+x^2/8-(x/8)*sqrt(1-6*x+x^2)) \\ Joerg Arndt, May 12 2013

CROSSREFS

Cf. A100299.

Sequence in context: A289603 A240049 A078418 * A027296 A179601 A151495

Adjacent sequences:  A100297 A100298 A100299 * A100301 A100302 A100303

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Nov 12 2004

STATUS

approved

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Last modified August 12 05:33 EDT 2020. Contains 336438 sequences. (Running on oeis4.)