%I A000672 M0326 N0122
%S A000672 1,1,1,1,2,2,4,6,11,18,37,66,135,265,552,1132,2410,5098,11020,23846,
%T A000672 52233,114796,254371,565734,1265579,2841632,6408674,14502229,32935002,
%U A000672 75021750,171404424,392658842,901842517,2076217086,4790669518,11077270335
%N A000672 Number of 3-valent trees (= boron trees or binary trees) with n nodes.
%C A000672 This can be described in 2 ways: (a) Trees with n nodes of valency <= 
               3, for n = 0,1,2,3,... (b) Trees with t = 2n+2 nodes of valency either 
               1 or 3 (implying that there are n nodes of valency 3 - the boron 
               atoms - and n+2 nodes of valency 1 - the hydrogen atoms), for t = 
               2,4,6,8,...
%D A000672 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000672 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000672 P. J. Cameron, Oligomorphic Permutation Groups, Cambridge; see Fig. 2 
               p. 35.
%D A000672 A. Cayley, On the analytical forms called trees, with application to 
               the theory of chemical combinations, Reports British Assoc. Advance. 
               Sci. 45 (1875), 257-305 = Math. Papers, Vol. 9, 427-460 (see p. 451).
%D A000672 S. J. Cyvin et al., Enumeration of constitutional isomers of polyenes, 
               J. Molec. Structure (Theochem), 357 (1995), 255-261.
%D A000672 R. C. Read, personal communication.
%H A000672 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Sequences realized by oligomorphic permutation groups</a>, J. Integ. 
               Seqs. Vol. 3 (2000), #00.1.5.
%H A000672 E. M. Rains and N. J. A. Sloane, <a href="http://www.cs.uwaterloo.ca/
               journals/JIS/index.html">On Cayley's Enumeration of Alkanes (or 4-Valent 
               Trees).</a>, J. Integer Sequences, Vol. 2 (1999), Article 99.1.1.
%H A000672 <a href="Sindx_Tra.html#trees">Index entries for sequences related to 
               trees</a>
%H A000672 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               TrivalentTree.html">Trivalent Tree</a>
%F A000672 Rains and Sloane give a g.f.
%F A000672 a(0)=a(1)=a(2)=1, a(n) = 2b(n+1) - b(n+2) + b((n+1)/2) - 2 C(1+b(n/3), 
               3) - Sum_{i=1..[(n-1)/2]} C(b(i), 2)b(n-2i) + Sum_{i=1..[n/3]} b(i) 
               Sum_{j=i..[(n-i)/2]} b(j)b(n-i-j), where b(x) = A001190(x) if x is 
               an integer, otherwise 0 (Cyvin et al.)
%e A000672 The 4 trees with 6 nodes are:
%e A000672 ._._._._._. . ._._._._. . ._._._._. . ._._._.
%e A000672 . . . . . . . . | . . . . . . | . . . . | |
%Y A000672 Equals A000673 + A000675. Cf. A052120, A000022, A000200, A000602.
%Y A000672 Sequence in context: A032237 A124346 A033961 this_sequence A115868 A103299 
               A154779
%Y A000672 Adjacent sequences: A000669 A000670 A000671 this_sequence A000673 A000674 
               A000675
%K A000672 nonn,easy,nice
%O A000672 0,5
%A A000672 N. J. A. Sloane (njas(AT)research.att.com).