%I A054581
%S A054581 1,1,1,2,5,12,39,136,529,2171,9368,41534,188942,874906,4115060,19602156,
%T A054581 94419351,459183768,2252217207,11130545494,55382155396,277255622646,
%U A054581 1395731021610,7061871805974,35896206800034,183241761631584
%N A054581 Number of unlabeled 2-trees with n nodes.
%C A054581 A 2-tree is recursively defined as follows: K_2 is a 2-tree and any 2-tree 
               on n+1 vertices is obtained by joining a vertex to a 2-clique in 
               a 2-tree on n vertices. Care is needed with the term 2-tree (and 
               k-tree in general) because it has at least two commonly used definitions.
%C A054581 A036361 gives the labeled version of this sequence, which has an easy 
               formula analagous to Cayley's formula for the number of trees.
%C A054581 Also, number of unlabeled 3-gonal 2-trees with n 3-gons.
%D A054581 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 
               1973, p. 76, t(x), (3.5.19).
%H A054581 <a href="Sindx_Tra.html#trees">Index entries for sequences related to 
               trees</a>
%H A054581 G. Labelle, C. Lamathe and P. Leroux, <a href="http://arXiv.org/abs/math.CO/
               0312424">Labeled and unlabeled enumeration of k-gonal 2-trees</a>
%e A054581 a(1)=a(2)=a(3)=1 because: K_2, K_3 are the only 2-trees on 2 and 3 nodes 
               and on 4 nodes, there is a also unique example obtained by joining 
               a triangle to K_3 along an edge (thus forming K_4\e). The two graphs 
               on 5 nodes are obtained by joining a triangle to K_4\e, either along 
               the shared edge or along one of the non-shared edges.
%Y A054581 Cf. A036361.
%Y A054581 Cf. A000272 (labeled trees), A036361 (labeled 2-trees), A036362 (labeled 
               3-trees), A036506 (labeled 4-trees), A000055 (unlabeled trees).
%Y A054581 Sequence in context: A050237 A050258 A051436 this_sequence A140440 A005664 
               A009739
%Y A054581 Adjacent sequences: A054578 A054579 A054580 this_sequence A054582 A054583 
               A054584
%K A054581 nonn,nice
%O A054581 1,4
%A A054581 Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 11 2000
%E A054581 Additional comments from Gordon Royle (gordon(AT)maths.uwa.edu.au), Dec 
               02 2002