%I A000014 M0320 N0118
%S A000014 0,1,1,0,1,1,2,2,4,5,10,14,26,42,78,132,249,445,842,1561,2988,5671,10981,
%T A000014 21209,41472,81181,160176,316749,629933,1256070,2515169,5049816,10172638,
%U A000014 20543579,41602425,84440886,171794492,350238175,715497037,1464407113
%N A000014 Number of series-reduced trees with n nodes.
%D A000014 F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like 
               Structures, Camb. 1998, p. 284.
%D A000014 D. G. Cantor, personal communication.
%D A000014 F. Harary, Graph Theory. Addison-Wesley, Reading, MA, 1969, p. 232.
%D A000014 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 
               1973, p. 62, Fig. 3.3.3.
%D A000014 F. Harary and G. Prins, The number of homeomorphically irreducible trees 
               and other species, Acta Math., 101 (1959), 141-162.
%D A000014 F. Harary, R. W. Robinson and A. J. Schwenk, Twenty-step algorithm for 
               determining the asymptotic number of trees of various species, J. 
               Austral. Math. Soc., Series A, 20 (1975), 483-503. Errata: Vol. A 
               41 (1986), p. 325.
%D A000014 J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 
               2004; p. 526.
%D A000014 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000014 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A000014 Christian G. Bower, <a href="b000014.txt">Table of n, a(n) for n = 0..500</
               a>
%H A000014 N. J. A. Sloane, <a href="a14.gif">Illustration of initial terms</a>
%H A000014 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Series-ReducedTree.html">Link to a section of The World of Mathematics.</
               a>
%H A000014 <a href="Sindx_Tra.html#trees">Index entries for sequences related to 
               trees</a>
%H A000014 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A000014 G.f.: A(x) = ((x-1)/x)*f(x) + ((1+x)/x^2)*g(x) - (1/x^2)*g(x)^2 where 
               f(x) is g.f. for A059123 and 1+g(x) is g.f. for A001678. [Harary 
               and E. M. Palmer, p. 62, Eq. (3.3.10) with extra -(1/x^2)*Hbar(x)^2 
               term which should be there according to eq.(3.3.14), p. 63, with 
               eq.(3.3.9)].
%p A000014 with (powseries): with (combstruct): n := 30: Order := n+3: sys := {B 
               = Prod(C,Z), S = Set(B,1 <= card), C = Union(Z,S)}:
%p A000014 G001678 := (convert(gfseries(sys,unlabeled,x) [S(x)], polynom)) * x^2: 
               G0temp := G001678 + x^2:
%p A000014 G059123 := G0temp / x + G0temp - (G0temp^2+eval(G0temp,x=x^2))/(2*x):
%p A000014 G000014 := ((x-1)/x) * G059123 + ((1+x)/x^2) * G0temp - (1/x^2) * G0temp^2:
%p A000014 A000014 := 0,seq(coeff(G000014,x^i),i=1..n); # from UlrSchimke(AT)aol.com
%Y A000014 Cf. A000055 (trees), A001678 (series-reduced planted trees), A007827 
               (series-reduced trees by leaves).
%Y A000014 Sequence in context: A147678 A127712 A032090 this_sequence A114851 A099364 
               A125951
%Y A000014 Adjacent sequences: A000011 A000012 A000013 this_sequence A000015 A000016 
               A000017
%K A000014 nonn,easy,core,nice
%O A000014 0,7
%A A000014 N. J. A. Sloane (njas(AT)research.att.com).
%E A000014 G.f. corrected by Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), 
               Jan 09 2001.