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%I #21 Jul 26 2022 12:46:42
%S 1,4,24,148,925,5838,37128,237576,1527867,9867000,63946740,415683216,
%T 2709186844,17697136408,115833872400,759517409424,4987999112007,
%U 32804320226580,216018805979760,1424151150922500,9398957079664845,62090203617715350,410536632908307360
%N Number of branches in all noncrossing rooted trees on n nodes on a circle.
%H Andrew Howroyd, <a href="/A045738/b045738.txt">Table of n, a(n) for n = 2..200</a>
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F a(n) = binomial(3n-3, n-2) - 2*binomial(3n-6, n-3).
%F G.f.: (2*g^3-4*g^2+2*g-1)/((1-3*g)*(g-1)^3) where g*(1-g)^2 = x. - _Mark van Hoeij_, Nov 10 2011
%F D-finite with recurrence +2*(2*n-1)*(n-2)*a(n) +(-43*n^2+169*n-160)*a(n-1) +4*(31*n^2-196*n+292)*a(n-2) -12*(3*n-13)*(3*n-14)*a(n-3)=0. - _R. J. Mathar_, Jul 26 2022
%o (PARI) a(n) = binomial(3*n-3, n-2) - 2*binomial(3*n-6, n-3); \\ _Andrew Howroyd_, Nov 12 2017
%o (PARI) \\ here b(n) is x^2 * g.f. of A006013.
%o b(n)={serreverse(x-2*x^2+x^3 + O(x^n))}
%o s(n)={(g->(2*g^3-4*g^2+2*g-1)/((1-3*g)*(g-1)^3))(b(n)) + O(x^n)}
%o Vec(s(25)) \\ _Andrew Howroyd_, Nov 12 2017
%K nonn
%O 2,2
%A _Emeric Deutsch_
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