

A121446


Number of ternary trees with n edges and such that the first leaf in the preorder traversal is at level 1.


1



3, 3, 10, 42, 198, 1001, 5304, 29070, 163438, 937365, 5462730, 32256120, 192565800, 1160346492, 7048030544, 43108428198, 265276342782, 1641229898525, 10202773534590, 63698396932170, 399223286267190, 2510857763851185, 15842014607109600, 100244747986099080
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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

Table of n, a(n) for n=1..24.
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(n1) for n >= 2.
a(1) = 3 and a(n) = (2/n)*binomial(3*n3, n1) 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).
Dfinite with recurrence 2*n*(2*n  3)*a(n)  3*(3*n  4)*(3*n  5)*a(n1) = 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*n3, n1) 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] (* JeanFrançois Alcover, Nov 28 2017 *)


CROSSREFS

Cf. A007226.
Column 1 of A121445.
Sequence in context: A107299 A298899 A205388 * A302196 A340598 A258193
Adjacent sequences: A121443 A121444 A121445 * A121447 A121448 A121449


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Jul 30 2006


STATUS

approved



