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%I #14 Aug 23 2023 08:43:25
%S 0,0,1,4,34,633,23368,1699012,237934760,64558137140,34126032806936,
%T 35513501049012952
%N Number of n-vertex labeled simple graphs containing a Hamiltonian path.
%C A path is Hamiltonian if it passes through every vertex exactly once.
%H F. Hüffner, <a href="https://github.com/falk-hueffner/tinygraph">tinygraph</a>, software for generating integer sequences based on graph properties, version 9766535.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Hamiltonian_path">Hamiltonian path</a>
%H Gus Wiseman, <a href="http://arxiv.org/abs/0709.0430">Enumeration of paths and cycles and e-coefficients of incomparability graphs</a>, arXiv:0709.0430 [math.CO], 2007.
%H Gus Wiseman, <a href="/A326206/a326206.png">The a(4) = 34 simple graphs containing a Hamiltonian path</a>
%F A006125(n) = a(n) + A326205(n).
%t Table[Length[Select[Subsets[Subsets[Range[n],{2}]],FindHamiltonianPath[Graph[Range[n],#]]!={}&]],{n,0,4}] (* Mathematica 10.2+ *)
%Y The unlabeled case is A057864.
%Y The directed case is A326214 (with loops) or A326217 (without loops).
%Y Simple graphs without a Hamiltonian path are A326205.
%Y Simple graphs with a Hamiltonian cycle are A326208.
%Y Cf. A003216, A006125, A057864, A283420.
%K nonn,more
%O 0,4
%A _Gus Wiseman_, Jun 14 2019
%E a(7)-a(11) added using tinygraph by _Falk Hüffner_, Jun 21 2019