cp's OEIS Frontend

A121446 Number of ternary trees with n edges and such that the first leaf in the preorder traversal is at level 1.

%I A121446 #43 Jul 31 2025 08:42:46
%S A121446 3,3,10,42,198,1001,5304,29070,163438,937365,5462730,32256120,
%T A121446 192565800,1160346492,7048030544,43108428198,265276342782,
%U A121446 1641229898525,10202773534590,63698396932170,399223286267190,2510857763851185,15842014607109600,100244747986099080
%N A121446 Number of ternary trees with n edges and such that the first leaf in the preorder traversal is at level 1.
%C A121446 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 A121446 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 A121446 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 A121446 a(n) = A007226(n-1) for n >= 2.
%F A121446 a(1) = 3 and a(n) = (2/n)*binomial(3*n-3, n-1) for n >= 2.
%F A121446 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 A121446 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 A121446 G.f.: 1-(1-(4*sin(arcsin((3^(3/2)*sqrt(x))/2)/3)^2)/3)^3. - _Vladimir Kruchinin_, Oct 04 2022
%F A121446 From _Peter Bala_, Jul 24 2025: (Start)
%F A121446 The g.f. A(x) = 3*x + 3*x^2 + 10*x^3 + ... satisfies the algebraic equation A(x)^3 - (3*x + 2)*A(x)^2 + (3*x^2 + 6*x + 1)*A(x) - (x^3 + 3*x^2 + 3*x) = 0.
%F A121446 1 + x/(1 - A(x)) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + ... is the g.f. of A001764.
%F A121446 The g.f. A(x) satisfies (and is uniquely determined by) the conditions [x^n] (A(x) - 1)^n = 3 for n >= 1. (End)
%e A121446 a(1) = 3 because we have the trees /, | and \.
%e A121446 a(2) = 3 because we have the trees /|, /\ and |\.
%p A121446 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 A121446 a[1] = 3; a[n_] := (2/n) Binomial[3 n - 3, n - 1];
%t A121446 Array[a, 22] (* _Jean-François Alcover_, Nov 28 2017 *)
%Y A121446 Cf. A007226, A001764, A006013, A006632.
%Y A121446 Column 1 of A121445.
%K A121446 nonn,easy
%O A121446 1,1
%A A121446 _Emeric Deutsch_, Jul 30 2006