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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 (list; graph; refs; listen; history; text; internal format)
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(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

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

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Last modified February 6 23:12 EST 2023. Contains 360111 sequences. (Running on oeis4.)