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%I #35 Jul 26 2022 16:22:28
%S 1,0,0,1,7,42,245,1428,8379,49588,296010,1781325,10798788,65900296,
%T 404565252,2496994136,15486165555,96464124648,603262881620,
%U 3786268349115,23842082904255,150586208376450,953736669989985
%N Number of noncrossing trees with n edges in which no border edges emanate from the root.
%H Andrew Howroyd, <a href="/A102594/b102594.txt">Table of n, a(n) for n = 0..200</a>
%H 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.
%H Emanuele Munarini, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Munarini/muna4.html">Shifting Property for Riordan, Sheffer and Connection Constants Matrices</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.
%H M. Noy, <a href="http://dx.doi.org/10.1016/S0012-365X(97)00121-0">Enumeration of noncrossing trees on a circle</a>, Discrete Math., 180, 301-313, 1998.
%F a(n) = 7/3*(n-1)*(n-2)*binomial(3*n, n)/(3*n-1)/(2*n+1)/(3*n-2) for n > 0; a(0)=1.
%F G.f.: g*(1+z-2*z*g), where g = 1+z*g^3 is the g.f. of the ternary numbers (A001764).
%F From _Karol A. Penson_, Mar 12 2018: (Start)
%F a(n+3) = 7*binomial(3*n+6, 2*n+6)/(2*n+7).
%F a(n+3) is the n-th moment of a signed function v(x) on (0,27/4), i.e., in Maple notation, a(n+3) = int(x^n*v(x) , x = 0..27/4), n = 0,1..., where v(x) = -sqrt(3)*x^(4/3)*(7*x^(1/3)*hypergeom([-5/6, -1/3, 8/3], [2/3, 4/3], 4*x/27))-3*hypergeom([-7/6, -2/3, 7/3], [1/3, 2/3], 4*x/27)))/(6*Pi). The function v(x) vanishes at x = 0 and x = 27/4. In addition it has one zero in the interval between x = 0 and x = 27/4. (End)
%F D-finite with recurrence 2*n*(2*n+1)*(n-3)*a(n) -3*(3*n-5)*(n-1)*(3*n-4)*a(n-1)=0. - _R. J. Mathar_, Jul 26 2022
%e a(2)=0 because in all the three noncrossing trees with 2 edges, namely, /_, /\ and _\, the root (=the top vertex) is incident with at least one border edge.
%p a:=n->7/3*(n-1)*(n-2)*binomial(3*n,n)/(3*n-1)/(2*n+1)/(3*n-2): 1,seq(a(n), n=1..25);
%t a[0] = a[3] = 1; a[n_] := 7*Binomial[3n-3, 2n+1]/(n-3); Table[a[n], {n, 0, 22}] (* _Jean-François Alcover_, Jan 21 2013 *)
%o (PARI) a(n) = if (n==0, 1, 7/3*(n-1)*(n-2)*binomial(3*n, n)/(3*n-1)/(2*n+1)/(3*n-2)); \\ _Michel Marcus_, Oct 26 2015
%o (PARI) Vec((g->g*(1+x-2*x*g))(1+serreverse(x/(1+x)^3 + O(x^30)))) \\ _Andrew Howroyd_, Nov 17 2017
%Y Column k=0 of A102593.
%Y Cf. A001764.
%K nonn
%O 0,5
%A _Emeric Deutsch_, Jan 22 2005
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