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%I #29 Jul 26 2022 12:45:17
%S 0,0,3,28,210,1470,9996,67032,446292,2960100,19594575,129585456,
%T 856703848,5663913528,37454912040,247778648880,1639890119016,
%U 10858731869160,71939098633185,476841658085100,3162310375905450
%N Number of nonroot branch nodes in all noncrossing rooted trees on n nodes on a circle.
%H Andrew Howroyd, <a href="/A045737/b045737.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) = 7*(n-1)*binomial(3n-6, n-4)/(2n-1).
%F G.f.: g^2*(2*g-3)/((1-3*g)*(g-1)^3) where g*(1-g)^2 = x. - _Mark van Hoeij_, Nov 10 2011
%F a(n) = Sum_{k=0..ceiling(n/2)-1} k*A101431(n, k). - _Andrew Howroyd_, Nov 17 2017
%F D-finite with recurrence +2*(2*n-1)*(n-4)*a(n) -3*(3*n-7)*(3*n-8)*a(n-1)=0. - _R. J. Mathar_, Jul 26 2022
%t Table[7(n-1) Binomial[3n-6,n-4]/(2n-1),{n,2,30}] (* _Harvey P. Dale_, Nov 01 2017 *)
%o (PARI) a(n) = 7*(n-1)*binomial(3*n-6, n-4)/(2*n-1); \\ _Andrew Howroyd_, Nov 11 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->g^2*(2*g-3)/((1-3*g)*(g-1)^3))(b(n)) + O(x^n)}
%o concat([0,0],Vec(s(25))) \\ _Andrew Howroyd_, Nov 11 2017
%Y Cf. A006013, A101431.
%K nonn
%O 2,3
%A _Emeric Deutsch_
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