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%I #41 Jul 26 2022 16:30:12
%S 1,0,1,5,25,130,700,3876,21945,126500,740025,4382625,26225628,
%T 158331880,963250600,5899491640,36345082425,225082957512,
%U 1400431689475,8749779798375,54874635255825,345329274848250,2179969531405680
%N Number of noncrossing trees with n edges and having degree of the root at least 2.
%C [a(n+2)]= [1,5,25,130,700,...] is the self-convolution 5th power of A001764. - _Philippe Deléham_, Nov 11 2009
%H Andrew Howroyd, <a href="/A102893/b102893.txt">Table of n, a(n) for n = 0..200</a>
%H David Bevan, Robert Brignall, Andrew Elvey Price and Jay Pantone, <a href="http://arxiv.org/abs/1711.10325">A structural characterisation of Av(1324) and new bounds on its growth rate</a>, arXiv preprint arXiv:1711.10325 [math.CO], 2017-2019.
%H Emanuele Munarini, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Munarini/muna4.pdf">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(0)=1; a(n) = 5*binomial(3n-1, n-2)/(3n-1) if n > 0.
%F G.f.: g - z*g^2, where g = 1 + z*g^3 is the g.f. of the ternary numbers (A001764).
%F a(n) = A001764(n) - A006013(n-1).
%F D-finite with recurrence 2*n*(2*n+1)*(n-2)*a(n) -3*(n-1)*(3*n-4)*(3*n-2)*a(n-1)=0. - _R. J. Mathar_, Feb 16 2018
%F a(n) ~ (5*3^(3*n + 1/2))/(36*4^n*n^(3/2)*sqrt(Pi)). - _Peter Luschny_, Aug 08 2020
%e a(2)=1 because among the noncrossing trees with 2 edges, namely /_, _\ and /\, only the last one has root degree >1.
%p a:=proc(n) if n=0 then 1 else 5*binomial(3*n-1,n-2)/(3*n-1) fi end:
%p seq(a(n), n=0..25);
%p # Recurrence:
%p a := proc(n) option remember; if n < 3 then return [1,0,1][n+1] fi;
%p (27*n^3 - 81*n^2 + 78*n - 24)*a(n - 1)/(4*n^3 - 6*n^2 - 4*n) end:
%p seq(a(n), n=0..23); # _Peter Luschny_, Aug 08 2020
%p alias(PS=ListTools:-PartialSums): A102893List := proc(m) local A, P, n;
%p A := [1,0]; P := [1]; for n from 1 to m - 2 do P := PS(PS([op(P), P[-1]]));
%p A := [op(A), P[-2]] od; A end: A102893List(23); # _Peter Luschny_, Mar 26 2022
%t a[0] = 1; a[n_] := 5*Binomial[3n-1, n-2]/(3n-1); Table[a[n], {n, 0, 22}] (* _Jean-François Alcover_, Mar 01 2018 *)
%o (PARI) a(n) = if(n<=1, n==0, 5*binomial(3*n-1, n-2)/(3*n-1)); \\ _Andrew Howroyd_, Nov 17 2017
%Y Column k=0 of A102892 and column k=0 of A102593.
%Y Cf. A001764, A006013.
%K nonn,easy
%O 0,4
%A _Emeric Deutsch_, Jan 16 2005
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