OFFSET
2,2
COMMENTS
Convolution of (1, A007297) with itself.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 2..200
P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204, 203-229, 1999.
FORMULA
a(n) = Sum_{k=2..2*n-4} 2*binomial(k-2, n-3)*binomial(3*n-5, 2*n-k-4)/(n-2) for n > 2. - Andrew Howroyd, Nov 12 2017
G.f.: g^2, where g satisfies g^3+g^2-3zg+2z^2=0, g(0)=0, or, in Maple notation, g := -1/3+(2/3)*sqrt(1+9*z)*sin((1/3)*arcsin((2+27*z+54*z^2)/2/(1+9*z)^(3/2))).
G.f.: (x+x*g)^2 where g satisfies g - g^2 = x*(1 + g)^3. - Andrew Howroyd, Nov 13 2017
a(n) ~ 2^(n-1) * 3^(3*n/2-9/4) / (sqrt(Pi)*n^(3/2)*sqrt(45+26*sqrt(3))). - Vaclav Kotesovec, Mar 17 2014
D-finite with recurrence n*(2*n-3)*(n-2)*a(n) +6*(9*n-10)*a(n-1) -12*(3*n-10)*(3*n-8)*(2*n-1)*a(n-2)=0. - R. J. Mathar, May 10 2018
EXAMPLE
a(3)=2 because among the four non-crossing graphs on the points A,B,C, the distinguished node A has degree equal to 1 only in the graphs {AB,BC} and {AC,BC}; in the other two graphs ({AB,AC} and {AB,BC,AC}) the node A has degree 2.
MATHEMATICA
terms = 21;
g[x_] = 0;
Do[g[x_] = g[x]^2 + x (1 + g[x])^3 + O[x]^(terms+2), {terms+2}];
Drop[CoefficientList[(x + x g[x])^2 + O[x]^(terms+2), x], 2] (* Jean-François Alcover, Oct 05 2011, updated Jul 29 2018 after Andrew Howroyd *)
PROG
(PARI) a(n)=if(n<3, n==2, sum(k=2, 2*n-4, 2*binomial(k-2, n-3)*binomial(3*n-5, 2*n-k-4))/(n-2)); \\ Andrew Howroyd, Nov 12 2017
(PARI) Vec((x+x*serreverse((x-x^2)/(1+x)^3 + O(x^25)))^2) \\ Andrew Howroyd, Nov 13 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Dec 28 2003
STATUS
approved