%I A059123
%S A059123 0,1,1,0,2,2,4,6,12,20,39,71,137,261,511,995,1974,3915,7841,15749,
%T A059123 31835,64540,131453,268498,550324,1130899,2330381,4813031,9963288,
%U A059123 20665781,42947715,89410092,186447559,389397778,814447067,1705775653
%N A059123 Number of homeomorphically irreducible rooted trees (also known as series-reduced 
               rooted trees, or rooted trees without nodes of degree 2) with n >
               = 1 nodes.
%D A059123 P. J. Cameron, Some treelike objects, Quart. J. Math. Oxford, 38 (1987), 
               155-183.
%D A059123 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 
               1973, p. 62, Eq. (3.3.9).
%H A059123 N. J. A. Sloane, <a href="b059123.txt">Table of n, a(n) for n = 0..500</
               a>
%H A059123 N. J. A. Sloane, <a href="a59123.jpeg">Illustration of initial terms</
               a>
%H A059123 <a href="Sindx_Tra.html#trees">Index entries for sequences related to 
               trees</a>
%H A059123 <a href="Sindx_Ro.html#rooted">Index entries for sequences related to 
               rooted trees</a>
%F A059123 G.f.: 1 + ((1+x)/x)*f(x) - (f(x)^2+f(x^2))/(2*x) where 1+f(x) is g.f. 
               for A001678 (homeomorphically irreducible planted trees by nodes).
%p A059123 with (powseries): with (combstruct): n := 30: Order := n+3: sys := {B 
               = Prod(C,Z), S = Set(B,1 <= card), C = Union(Z,S)}:
%p A059123 G001678 := (convert(gfseries(sys,unlabeled,x) [S(x)], polynom)) * x^2: 
               G0temp := G001678 + x^2:
%p A059123 G059123 := G0temp / x + G0temp - (G0temp^2+eval(G0temp,x=x^2))/(2*x): 
               A059123 := 0,seq(coeff(G059123,x^i),i=1..n); # from UlrSchimke(AT)aol.com
%Y A059123 Apart from initial term, same as A001679.
%Y A059123 Cf. A000055 (trees by nodes), A000014 (homeomorphically irreducible trees 
               by nodes), A000669 (homeomorphically irreducible planted trees by 
               leaves), A000081 (rooted trees by nodes).
%Y A059123 Sequence in context: A129860 A028408 A037163 this_sequence A001679 A030435 
               A063886
%Y A059123 Adjacent sequences: A059120 A059121 A059122 this_sequence A059124 A059125 
               A059126
%K A059123 nonn,easy,nice
%O A059123 0,5
%A A059123 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Jan 09 2001.