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A089436
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Number of non-crossing connected graphs on n nodes on a circle in which a fixed (distinguished) node has degree one.
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4
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1, 2, 9, 54, 374, 2820, 22485, 186494, 1592778, 13914108, 123750874, 1116809628, 10201516332, 94140605832, 876332565837, 8219124900558, 77594375595266, 736785675010380, 7031930543228910, 67420537625021460, 649070964647075700
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OFFSET
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2,2
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COMMENTS
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Convolution of (1, A007297) with itself.
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LINKS
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FORMULA
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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
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EXAMPLE
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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.
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MATHEMATICA
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terms = 21;
g[x_] = 0;
Do[g[x_] = g[x]^2 + x (1 + g[x])^3 + O[x]^(terms+2), {terms+2}];
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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