%I A101307
%S A101307 1,1,1,3,2,7,6,1,18,18,6,47,59,24,2,129,188,96,16,362,605,369,90,4,1038,
%T A101307 1948,1395,436,45,3022,6305,5164,1981,315,9,8917,20460,18885,8568,1830,
%U A101307 126,26600,66585,68352,35818,9565,1071,21,80098,217186,245497,145796
%N A101307 Triangle read by rows: T(n,k) is the number of ordered trees having n 
               edges and k branches of length 2.
%C A101307 Row n has 1+floor(n/2) terms (n>=0). Row sums are the Catalan numbers 
               (A000108). Column k=0 yields A101308. T(2n,n)=A001006(n-1) (n>0) 
               (the Motzkin numbers).
%D A101307 E. Deutsch, Ordered trees with prescribed root degrees, node degrees 
               and branch lengths, Discrete Math., 282, 2004, 89-94.
%D A101307 J. Riordan, Enumeration of plane trees by branches and endpoints, J. 
               Comb. Theory (A) 19, 1975, 214-222.
%F A101307 G.f.=G=G(t, z) satisfies G=1+P+PG(G-1), where P= z/(1-z)+(t-1)z^2 (for 
               the explicit form see the Maple program).
%e A101307 Triangle begins:
%e A101307 1;
%e A101307 1,1;
%e A101307 3,2;
%e A101307 7,6,1;
%e A101307 18,18,6;
%p A101307 G:=(1+t*z^2-z^2+z^3-t*z^3-sqrt((1+t*z^2-z^2+z^3-t*z^3)*(1-4*z+3*z^2-3*t*z^2-3*z^3+3*t*z^3)))/
               2/z/(1-z+t*z+z^2-t*z^2): Gserz:=simplify(series(G,z=0,16)): for n 
               from 1 to 14 do P[n]:=sort(coeff(Gserz,z^n)) od: for n from 1 to 
               14 do seq(coeff(t*P[n],t^k),k=1..1+floor(n/2)) od; # yields the sequence 
               in triangular form
%Y A101307 Cf. A000108, A101308, A001006.
%Y A101307 Sequence in context: A033318 A093780 A161749 this_sequence A096899 A154448 
               A099896
%Y A101307 Adjacent sequences: A101304 A101305 A101306 this_sequence A101308 A101309 
               A101310
%K A101307 nonn,tabf
%O A101307 1,4
%A A101307 Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 22 2004