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A100299 Number of dissections of a convex n-gon by nonintersecting diagonals into an even number of regions. 2
0, 2, 5, 23, 98, 452, 2139, 10397, 51524, 259430, 1323361, 6824435, 35519686, 186346760, 984400759, 5231789177, 27954506504, 150079713482, 809181079293, 4379654830223, 23787413800490, 129607968854732, 708230837732435, 3880366912218773, 21312485647242828, 117321536967959342 (list; graph; refs; listen; history; text; internal format)
OFFSET
3,2
LINKS
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-2)/2)} C(n-3, 2*k-1)*C(n+2*k-2, 2*k-1)/(2*k).
G.f.: x*(1 -2*x -7*x^2 -(1+x)*sqrt(1-6*x+x^2))/(8*(1+x)).
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
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
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
EXAMPLE
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}.
MAPLE
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);
MATHEMATICA
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 *)
PROG
(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
(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
(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
(SageMath)
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))
[A100299(n) for n in range(3, 41)] # G. C. Greubel, Feb 05 2023
CROSSREFS
Cf. A100300.
Sequence in context: A290887 A219889 A369834 * A371308 A038833 A279819
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Nov 12 2004
STATUS
approved

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Last modified April 16 00:59 EDT 2024. Contains 371696 sequences. (Running on oeis4.)