

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, 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
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OFFSET

1,1


COMMENTS



LINKS



FORMULA

G.f.=G=G(t,z)=1/[1t(h1z)/(h1)]1, where h=1+zh^3=2sin(arcsin(sqrt(27z/4))/3)/sqrt(3z).


EXAMPLE

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;


MAPLE

h:=2/sqrt(3*z)*sin(arcsin(sqrt(27*z/4))/3): G:=rationalize(1/(1t*(h1z)/(h1)))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


CROSSREFS



KEYWORD

nonn,tabf


AUTHOR



EXTENSIONS



STATUS

approved



