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

1,1

Comments

Sum of terms in row n is A001764(n+1). T(n,1)=A121446(n) Sum(k*T(n,k),k=1..n)=A121447(n).

Links

Table of n, a(n) for n=1..43.

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).

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/(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

Crossrefs

Cf. A001764, A121446, A121447.

Sequence in context: A232281 A223743 A117783 * A327798 A121074 A121072

Adjacent sequences: A121442 A121443 A121444 * A121446 A121447 A121448

Keyword

nonn,tabf

Author

Emeric Deutsch, Jul 30 2006

Extensions

Keyword tabl changed to tabf by Michel Marcus, Apr 09 2013

Status

approved