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%I #23 Feb 09 2023 21:57:43
%S 0,2,5,23,98,452,2139,10397,51524,259430,1323361,6824435,35519686,
%T 186346760,984400759,5231789177,27954506504,150079713482,809181079293,
%U 4379654830223,23787413800490,129607968854732,708230837732435,3880366912218773,21312485647242828,117321536967959342
%N Number of dissections of a convex n-gon by nonintersecting diagonals into an even number of regions.
%H Vincenzo Librandi, <a href="/A100299/b100299.txt">Table of n, a(n) for n = 3..200</a>
%H P. Flajolet and M. Noy, <a href="http://dx.doi.org/10.1016/S0012-365X(98)00372-0">Analytic combinatorics of non-crossing configurations</a>, Discrete Math., 204, 203-229, 1999.
%F a(n) = Sum_{k=1..floor((n-2)/2)} C(n-3, 2*k-1)*C(n+2*k-2, 2*k-1)/(2*k).
%F G.f.: x*(1 -2*x -7*x^2 -(1+x)*sqrt(1-6*x+x^2))/(8*(1+x)).
%F Recurrence (for n>4): (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). - _Vaclav Kotesovec_, Oct 17 2012
%F Asymptotic: a(n) ~ sqrt(3*sqrt(2)-4)*(3+2*sqrt(2))^(n-1) / (8*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 17 2012
%F D-finite with recurrence (n-1)*a(n) = (4*n-11)*a(n-1) +5*(2*n-7)*a(n-2) +(4*n-17)*a(n-3) -(n-6)*a(n-4). - _R. J. Mathar_, Jul 26 2022
%e a(5)=5 because for a convex pentagon ABCDE we obtain dissections with an even number of regions by one of the following sets of diagonals: {AC}, {BD}, {CE}, {DA} and {EB}.
%p a:=n->sum(binomial(n-3,2*k-1)*binomial(n+2*k-2,2*k-1)/2/k, k=1..floor((n-2)/2)): seq(a(n),n=3..33);
%t Take[CoefficientList[Series[x*(1-2*x-7*x^2-(1+x)*Sqrt[1-6*x +x^2])/(8*(1+x)), {x, 0, 20}], x],{4,-1}] (* _Vaclav Kotesovec_, Oct 17 2012 *)
%o (PARI) my(x='x+O('x^66)); concat([0], Vec(x*(1-2*x-7*x^2-(1+x)*sqrt(1-6*x+x^2))/(8*(1+x)))) \\ _Joerg Arndt_, May 12 2013
%o (PARI) a(n) = sum(k=1, (n-2)\2, binomial(n-3, 2*k-1)*binomial(n+2*k-2, 2*k-1)/(2*k)); \\ _Altug Alkan_, Oct 26 2015
%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 40); [0] cat Coefficients(R!( x*(1 -2*x -7*x^2 -(1+x)*Sqrt(1-6*x+x^2))/(8*(1+x)) )); // _G. C. Greubel_, Feb 05 2023
%o (SageMath)
%o def A100299(n): return sum( binomial(n-3, 2*k-1)*binomial(n+2*k-2, 2*k-1)/(2*k) for k in range(1,(n//2)+1))
%o [A100299(n) for n in range(3,41)] # _G. C. Greubel_, Feb 05 2023
%Y Cf. A100300.
%K nonn
%O 3,2
%A _Emeric Deutsch_, Nov 12 2004
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