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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.
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The unlabeled case is Unlabeled digraphs not containing a Hamiltonian path are A326224.
Unlabeled digraphs without a Hamiltonian path are counted by A326224.
The undirected unlabeled case is 15aboveA326224.
The undirected case is A326238.
Digraphs (without loops) containing a Hamiltonian path are counted by A326217.
Digraphs (without loops) not containing a Hamiltonian cycle are counted by A326218.
Number of labeled n-vertex digraphs (without loops) not containing a (directed) Hamiltonian path.