%I
%S 1,1,1,1,3,2,8,5,1,24,15,3,75,46,10,1,243,148,34,4,808,489,116,16,1,
%T 2742,1652,402,61,5,9458,5678,1408,228,23,1,33062,19792,4982,847,97,6,
%U 116868,69798,17783,3138,393,31,1,417022,248577,63967,11627,1557,143,7
%N Triangle read by rows: T(n,k) is the number of ordered trees with n edges having k evenlength branches starting at the root (0<=k<=n).
%C Row n has 1+floor(n/2) terms. Row sums are the Catalan numbers (A000108). T(n,0)=A000958(n1). Sum(k*T(n,k),k=0..floor(n/2))=A127540(n1).
%F G.f.=(1+z)/(1+zz*Ctz^2*C), where C =[1sqrt(14z)]/(2z) is the Catalan function.
%e T(2,0)=1 because we have the tree /\.
%e Triangle starts:
%e 1;
%e 1;
%e 1,1;
%e 3,2;
%e 8,5,1;
%e 24,15,3;
%p C:=(1sqrt(14*z))/2/z: G:=(1+z)/(1+zz*Ct*z^2*C): Gser:=simplify(series(G,z=0,17)): for n from 0 to 14 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 14 do seq(coeff(P[n],t,j),j=0..floor(n/2)) od; # yields sequence in triangular form
%Y Cf. A000108, A000958, A127538, A127540.
%K nonn,tabf
%O 0,5
%A _Emeric Deutsch_, Mar 01 2007
