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A065065 Number of noncrossing connected graphs with nodes on a circle having n edges.

%I A065065 #26 Jul 26 2022 12:52:58
%S A065065 1,3,13,64,341,1913,11132,66573,406653,2526351,15913347,101396034,
%T A065065 652378120,4232439734,27657380019,181872596607,1202641671293,
%U A065065 7991878198287,53343146808137,357464739709920,2404073823950915
%N A065065 Number of noncrossing connected graphs with nodes on a circle having n edges.
%H A065065 Andrew Howroyd, <a href="/A065065/b065065.txt">Table of n, a(n) for n = 1..200</a>
%H A065065 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 A065065 a(n) = Sum_{k=ceiling((n+3)/2)..n+1} binomial(3*k-3,n+k)*binomial(n-1,n-k+1)/(k-1).
%F A065065 G.f. satisfies: A(x) = x*(1+3*A(x)+4*A(x)^2+A(x)^3). - _Vladimir Kruchinin_, Nov 12 2014
%F A065065 a(n) = Sum_{m=n..2*n-2} A127537(m,n). - _Andrew Howroyd_, Nov 12 2017
%F A065065 D-finite with recurrence 8*n*(2*n+1)*a(n) +2*(-46*n^2+55*n-18)*a(n-1) +6*(-30*n^2+60*n-7)*a(n-2) +2*(n-3)*(28*n-163)*a(n-3) +93*(n-3)*(n-4)*a(n-4)=0. - _R. J. Mathar_, Jul 26 2022
%e A065065 a(3)=13 because we have 1 triangle on 3 nodes and 12 non-crossing trees on 4 nodes.
%p A065065 A065065 := n-> sum(binomial(3*k-3,n+k)*binomial(n-1,n-k+1)/(k-1),k=ceil((n+3)/2)..n+1);
%t A065065 terms = 21;
%t A065065 A[_] = 0;
%t A065065 Do[A[x_] = x (1 + 3 A[x] + 4 A[x]^2 + A[x]^3) + O[x]^(terms+1), {terms+1}];
%t A065065 CoefficientList[A[x]/x, x] (* _Jean-François Alcover_, Jul 29 2018, after _Vladimir Kruchinin_ *)
%o A065065 (PARI) a(n)=sum(k=ceil((n+3)/2), n+1, binomial(3*k-3,n+k)*binomial(n-1,n-k+1)/(k-1)); \\ _Andrew Howroyd_, Nov 12 2017
%o A065065 (PARI) Vec(serreverse(x/(1+3*x+4*x^2+x^3) + O(x^20))) \\ _Andrew Howroyd_, Nov 12 2017
%Y A065065 Cf. A007297, A127537.
%K A065065 nonn
%O A065065 1,2
%A A065065 _Emeric Deutsch_, Nov 06 2001