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<x>:=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
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Nov 12 2004
STATUS
approved