%I A101276
%S A101276 1,0,1,1,0,1,1,2,0,2,2,2,6,0,4,3,8,6,16,0,9,6,14,30,16,45,0,21,11,36,54,
%T A101276 106,45,126,0,51,22,74,168,196,360,126,357,0,127,43,173,372,706,675,
%U A101276 1197,357,1016,0,323,87,378,981,1636,2775,2268,3913,1016,2907,0,835,176
%N A101276 Triangle read by rows: T(n,k) is the number of ordered trees having n 
               edges and k branches of length 1.
%C A101276 Row n has n+1 terms (n>=0). Row sums are the Catalan numbers (A000108). 
               Column 0 yields A026418. T(n,n)=A001006(n-1) (n>0) (the Motzkin numbers).
%D A101276 E. Deutsch, Ordered trees with prescribed root degrees, node degrees 
               and branch lengths, Discrete Math., 282, 2004, 89-94.
%D A101276 J. Riordan, Enumeration of plane trees by branches and endpoints, J. 
               Comb. Theory (A) 19, 1975, 214-222.
%F A101276 G.f. G=G(t, z) satisfies z(t+z-tz)G^2-(1-z+tz+z^2-tz^2)G+1-z+tz+z^2-tz^2=0.
%e A101276 T(3,1)=2 because we have the tree with three edges hanging from the root 
               and the tree with one edge hanging from the root at the end of which 
               two edges are hanging.
%p A101276 G := 1/2/(-z^2+t*z^2-t*z)*(-1+z-t*z-z^2+t*z^2+sqrt(1-3*t^2*z^2-8*t*z^3+6*t^2*z^3+6*z^4*t-3*t^2*z^4-2*t*z-z^2-\
               3*z^4+2*z^3-2*z+4*t*z^2)): Gser:=simplify(series(G,z=0,13)): P[0]:=1: 
               for n from 1 to 11 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 11 
               do seq(coeff(t*P[n],t^k),k=1..n+1) od; # yields the sequence in triangular 
               form
%Y A101276 Cf. A000108, A000106, A026418.
%Y A101276 Sequence in context: A161872 A036461 A063088 this_sequence A103863 A166395 
               A061199
%Y A101276 Adjacent sequences: A101273 A101274 A101275 this_sequence A101277 A101278 
               A101279
%K A101276 nonn,tabl
%O A101276 0,8
%A A101276 Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 20 2004