OFFSET
0,2
COMMENTS
a(n) is the number of graphs with loops on n unlabeled vertices with every connected component having no more edges than vertices. - Andrew Howroyd, Feb 02 2024
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..500
FORMULA
Partial sums of A369200.
Euler transform of A369289. - Andrew Howroyd, Feb 02 2024
EXAMPLE
The a(0) = 1 through a(3) = 12 loop-graphs (loops shown as singletons):
{} {} {} {}
{{1}} {{1}} {{1}}
{{1,2}} {{1,2}}
{{1},{2}} {{1},{2}}
{{1},{1,2}} {{1},{1,2}}
{{1},{2,3}}
{{1,2},{1,3}}
{{1},{2},{3}}
{{1},{2},{1,3}}
{{1},{1,2},{1,3}}
{{1},{1,2},{2,3}}
{{1,2},{1,3},{2,3}}
MATHEMATICA
brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{(Union@@m)[[i]], p[[i]]}, {i, Length[p]}])], {p, Permutations[Range[Length[Union@@m]]]}]]];
Table[Length[Union[brute /@ Select[Subsets[Subsets[Range[n], {1, 2}]], Length[Select[Tuples[#], UnsameQ@@#&]]!=0&]]], {n, 0, 4}]
CROSSREFS
The covering case is A369200.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 22 2024
EXTENSIONS
a(7) onwards from Andrew Howroyd, Feb 02 2024
STATUS
approved