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COMMENTS
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It is observed by Gus Wiseman in A368596 and A368730 that this sequence appears to be the complement of those sequences. If this is the case, then a(n) is the number of labeled graphs with loops allowed in which each connected component has an equal number of vertices and edges and the conjectured formula holds. Terms for n >= 9 are expected to be 167341283, 4191140394, 116425416531, ... - Andrew Howroyd, Jan 10 2024
An equivalent conjecture is that a(n) is the number of loop-graphs with n vertices and n edges such that it is possible to choose a different vertex from each edge. I call these graphs choosable. For example, the a(3) = 17 choosable loop-graphs are the following (loops shown as singletons):
{{1},{2},{3}} {{1},{2},{1,3}} {{1},{1,2},{1,3}} {{1,2},{1,3},{2,3}}
{{1},{2},{2,3}} {{1},{1,2},{2,3}}
{{1},{3},{1,2}} {{1},{1,3},{2,3}}
{{1},{3},{2,3}} {{2},{1,2},{1,3}}
{{2},{3},{1,2}} {{2},{1,2},{2,3}}
{{2},{3},{1,3}} {{2},{1,3},{2,3}}
{{3},{1,2},{1,3}}
{{3},{1,2},{2,3}}
{{3},{1,3},{2,3}}
(End)
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