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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_{k=1..floor((n-1)/2)} C(n-3, 2*k-2)*C(n+2*k-3, 2*k-2)/(2*k-1). G.f.: (1/8)*( z + z^2 - z*sqrt(1-6*z+z^2) - 4*z^2/(1+z) ). (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 From Vladeta Jovovic, Nov 15 2004: (Start) a(n) = (A001003(n-2) - (-1)^n)/2. a(n) = A100299(n) - (-1)^n. (End) 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 From G. C. Greubel, Feb 04 2023: (Start) a(n) = ( (10*(n-5)^2 + 37*(n-5) + 30)*a(n-1) + (10*(n-5)^2 + 43*(n-5) + 42)*a(n-2) - (n-5)*(2*n-5)*a(n-3) )/((n-1)*(2*n-7)), with a(3) = a(4) = 1, a(5) = 6. a(n) = Hypergeometric4F3([n/2, (n+1)/2, (3-n)/2, (4-n)/2], [1/2, 1, 3/2], 1). (End) 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)); seq(a(n), n=3..40); 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, 40}], x], {4, -1}] (* Vaclav Kotesovec, Oct 17 2012 *) PROG (PARI) my(x='x+O('x^66)); Vec(x*((1+x)^2 -(1+x)*sqrt(1-6*x+x^2) -4*x)/(8*(1+x))) \\ Joerg Arndt, May 12 2013 (Magma) R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( x*((1+x)^2 -(1+x)*Sqrt(1-6*x+x^2) -4*x)/(8*(1+x)) )); // G. C. Greubel, Feb 04 2023 (SageMath) def A100300(n): return sum(binomial(n-3, 2*k)*binomial(n+2*k-1, 2*k)/(2*k+1) for k in range((n-3)//2 +1)) [A100300(n) for n in range(3, 41)] # G. C. Greubel, Feb 04 2023 CROSSREFS Cf. A001003, A100299. Sequence in context: A349834 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 March 26 02:36 EDT 2023. Contains 361529 sequences. (Running on oeis4.)