A121445 Triangle read by rows: T(n,k) is the number of ternary trees with n edges and such that the first leaf in the preorder traversal is at level k (1<=k<=n). 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.
3, 3, 9, 10, 18, 27, 42, 69, 81, 81, 198, 312, 351, 324, 243, 1001, 1540, 1701, 1566, 1215, 729, 5304, 8034, 8784, 8100, 6480, 4374, 2187, 29070, 43554, 47313, 43713, 35640, 25515, 15309, 6561, 163438, 242896, 262684, 243108, 200745, 148716, 96957
Offset: 1
Examples
T(1,1)=3 because we have the trees /, | and \. T(2,1)=3 because we have the trees /|, /\ and |\. Triangle starts: 3; 3,9; 10,18,27; 42,69,81; 198,312,351,324,243;
Programs
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Maple
h:=2/sqrt(3*z)*sin(arcsin(sqrt(27*z/4))/3): G:=rationalize(1/(1-t*(h-1-z)/(h-1)))-1: Gser:=simplify(series(G,z=0,18)): for n from 1 to 10 do P[n]:=sort(expand(coeff(Gser,z^n))) od: for n from 1 to 10 do seq(coeff(P[n],t,j),j=1..n) od; # yields sequence in triangular form
Formula
G.f.=G=G(t,z)=1/[1-t(h-1-z)/(h-1)]-1, where h=1+zh^3=2sin(arcsin(sqrt(27z/4))/3)/sqrt(3z).
Extensions
Keyword tabl changed to tabf by Michel Marcus, Apr 09 2013
Comments