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%I #36 Oct 04 2022 20:31:11
%S 3,3,10,42,198,1001,5304,29070,163438,937365,5462730,32256120,
%T 192565800,1160346492,7048030544,43108428198,265276342782,
%U 1641229898525,10202773534590,63698396932170,399223286267190,2510857763851185,15842014607109600,100244747986099080
%N Number of ternary trees with n edges and such that the first leaf in the preorder traversal is at level 1.
%C A ternary tree is a rooted tree in which each vertex has at most three children and each child of a vertex is designated as its left or middle or right child.
%H Ira Gessel and Guoce Xin, <a href="https://arxiv.org/abs/math/0505217">The generating function of ternary trees and continued fractions</a>, arXiv:math/0505217 [math.CO], 2005.
%H Ira Gessel and Guoce Xin, <a href="https://doi.org/10.37236/1079">The generating function of ternary trees and continued fractions</a>, Electronic Journal of Combinatorics, 13(1) (2006), #R53.
%F a(n) = A007226(n-1) for n >= 2.
%F a(1) = 3 and a(n) = (2/n)*binomial(3*n-3, n-1) for n >= 2.
%F G.f.: (h - 1 - z)/(h - 1), where h = 1 + z*h^3 = 2*sin(arcsin(sqrt(27*z/4))/3)/sqrt(3*z).
%F D-finite with recurrence 2*n*(2*n - 3)*a(n) - 3*(3*n - 4)*(3*n - 5)*a(n-1) = 0 for n >= 3. - _R. J. Mathar_, Jun 22 2016
%F G.f.: 1-(1-(4*sin(arcsin((3^(3/2)*sqrt(x))/2)/3)^2)/3)^3. - _Vladimir Kruchinin_, Oct 04 2022
%e a(1) = 3 because we have the trees /, | and \.
%e a(2) = 3 because we have the trees /|, /\ and |\.
%p a:=proc(n) if n=1 then 3 else (2/n)*binomial(3*n-3,n-1) fi end: seq(a(n),n=1..25);
%t a[1] = 3; a[n_] := (2/n) Binomial[3 n - 3, n - 1];
%t Array[a, 22] (* _Jean-François Alcover_, Nov 28 2017 *)
%Y Cf. A007226.
%Y Column 1 of A121445.
%K nonn
%O 1,1
%A _Emeric Deutsch_, Jul 30 2006