A121445 - OEIS

%I #7 Apr 09 2013 09:41:53

%S 3,3,9,10,18,27,42,69,81,81,198,312,351,324,243,1001,1540,1701,1566,

%T 1215,729,5304,8034,8784,8100,6480,4374,2187,29070,43554,47313,43713,

%U 35640,25515,15309,6561,163438,242896,262684,243108,200745,148716,96957

%N 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.

%C 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).

%F 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).

%e T(1,1)=3 because we have the trees /, | and \.

%e T(2,1)=3 because we have the trees /|, /\ and |\.

%e Triangle starts:

%e 3;

%e 3,9;

%e 10,18,27;

%e 42,69,81;

%e 198,312,351,324,243;

%p 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

%Y Cf. A001764, A121446, A121447.

%K nonn,tabf

%O 1,1

%A _Emeric Deutsch_, Jul 30 2006

%E Keyword tabl changed to tabf by _Michel Marcus_, Apr 09 2013