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%I #68 Mar 02 2024 11:59:54
%S 0,1,1,1,2,5,12,39,136,529,2171,9368,41534,188942,874906,4115060,
%T 19602156,94419351,459183768,2252217207,11130545494,55382155396,
%U 277255622646,1395731021610,7061871805974,35896206800034,183241761631584
%N Number of unlabeled 2-trees with n nodes.
%C 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 A036361 gives the labeled version of this sequence, which has an easy formula analogous to Cayley's formula for the number of trees.
%C Also, number of unlabeled 3-gonal 2-trees with n 3-gons.
%D Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 327-328.
%D F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 76, t(x), (3.5.19).
%H Allan Bickle, <a href="https://doi.org/10.20429/tag.2024.000105">A Survey of Maximal k-degenerate Graphs and k-Trees</a>, Theory and Applications of Graphs 0 1 (2024) Article 5.
%H T. Fowler, I. Gessel, G. Labelle, and P. Leroux, <a href="https://doi.org/10.1006/aama.2001.0771">The specification of 2-trees</a>, Adv. Appl. Math. 28 (2) (2002) 145-168, Table 1.
%H Nick Early, Anaëlle Pfister, and Bernd Sturmfels, <a href="https://arxiv.org/abs/2402.03065">Minimal Kinematics on M_{0,n}</a>, arXiv:2402.03065 [math.AG], 2024.
%H Andrew Gainer-Dewar, <a href="https://doi.org/10.37236/2615">Gamma-Species and the Enumeration of k-Trees</a>, Electronic Journal of Combinatorics, Volume 19 (2012), #P45. - From _N. J. A. Sloane_, Dec 15 2012
%H Gilbert Labelle, Cédric Lamathe, and Pierre Leroux, <a href="https://arxiv.org/abs/math/0312424">Labeled and unlabeled enumeration of k-gonal 2-trees</a>, arXiv:math/0312424 [math.CO], 2003.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/k-Tree.html">k-Tree</a>.
%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%e a(1)=0 because K_1 is not a 2-tree;
%e a(2)=a(3)=1 because K_2 and K_3 are the only 2-trees on those sizes.
%e a(4)=1 because there is a 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 Column k=3 of A340811, column k=2 of A370770.
%Y Cf. A000272 (labeled trees), A036361 (labeled 2-trees), A036362 (labeled 3-trees), A036506 (labeled 4-trees), A000055 (unlabeled trees).
%K nonn,nice
%O 1,5
%A _Vladeta Jovovic_, Apr 11 2000
%E Additional comments from _Gordon F. Royle_, Dec 02 2002
%E Missing initial term 0 inserted by _Brendan McKay_, Aug 07 2023
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