OFFSET
1,1
COMMENTS
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.
LINKS
Ira Gessel and Guoce Xin, The generating function of ternary trees and continued fractions, arXiv:math/0505217 [math.CO], 2005.
Ira Gessel and Guoce Xin, The generating function of ternary trees and continued fractions, Electronic Journal of Combinatorics, 13(1) (2006), #R53.
FORMULA
a(n) = A007226(n-1) for n >= 2.
a(1) = 3 and a(n) = (2/n)*binomial(3*n-3, n-1) for n >= 2.
G.f.: (h - 1 - z)/(h - 1), where h = 1 + z*h^3 = 2*sin(arcsin(sqrt(27*z/4))/3)/sqrt(3*z).
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
G.f.: 1-(1-(4*sin(arcsin((3^(3/2)*sqrt(x))/2)/3)^2)/3)^3. - Vladimir Kruchinin, Oct 04 2022
EXAMPLE
a(1) = 3 because we have the trees /, | and \.
a(2) = 3 because we have the trees /|, /\ and |\.
MAPLE
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);
MATHEMATICA
a[1] = 3; a[n_] := (2/n) Binomial[3 n - 3, n - 1];
Array[a, 22] (* Jean-François Alcover, Nov 28 2017 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Jul 30 2006
STATUS
approved