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%I #2 Mar 30 2012 17:36:12
%S 1,0,1,1,0,1,1,3,0,1,3,5,5,0,1,7,18,9,7,0,1,20,52,37,13,9,0,1,59,168,
%T 113,60,17,11,0,1,184,546,388,190,87,21,13,0,1,593,1826,1313,688,283,
%U 118,25,15,0,1,1964,6211,4545,2408,1076,392,153,29,17,0,1,6642,21459
%N Triangle read by rows: T(n,k) is the number of ordered trees with n edges and having k branches of length 1 starting from the root (0<=k<=n).
%C Row sums are the Catalan numbers (A000108). T(n,0)=A030238(n-2) for n>=2. Sum(k*T(n,k),k=0..n)=A026012(n-1) for n>=1.
%F G.f.= 1/(1-tzC+tz^2*C-z^2*C), where C=[1-sqrt(1-4z)]/(2z) is the Catalan function.
%e Triangle starts:
%e 1;
%e 0,1;
%e 1,0,1;
%e 1,3,0,1;
%e 3,5,5,0,1;
%e 7,18,9,7,0,1;
%p C:=(1-sqrt(1-4*z))/2/z: G:=1/(1-t*z*C+t*z^2*C-z^2*C): Gser:=simplify(series(G,z=0,15)): for n from 0 to 12 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 12 do seq(coeff(P[n],t,j),j=0..n) od; # yields sequence in triangular form
%Y Cf. A000108, A030238, A026012.
%K nonn,tabl
%O 0,8
%A _Emeric Deutsch_, Mar 01 2007
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