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A322698 Number of regular graphs with half-edges on n labeled vertices.

%I A322698 #12 Mar 09 2020 21:19:33
%S A322698 1,2,4,10,40,278,3554,84590,3776280,317806466,50710452574,
%T A322698 15414839551538,8964708979273634,10008446308186072290,
%U A322698 21518891146915893435358,89320970210116481106835986,717558285660687970023516336792,11176382741327158622885664697124082,338202509574712032788035618665293979610
%N A322698 Number of regular graphs with half-edges on n labeled vertices.
%C A322698 A graph is regular if all vertices have the same degree. A half-edge is like a loop except it only adds 1 to the degree of its vertex.
%H A322698 Wikipedia, <a href="https://en.wikipedia.org/wiki/Regular_graph">Regular graph</a>
%e A322698 The a(3) = 10 edge sets:
%e A322698   {}
%e A322698   {{1},{2,3}}
%e A322698   {{3},{1,2}}
%e A322698   {{2},{1,3}}
%e A322698   {{1},{2},{3}}
%e A322698   {{1,2},{1,3},{2,3}}
%e A322698   {{1},{3},{1,2},{2,3}}
%e A322698   {{1},{2},{1,3},{2,3}}
%e A322698   {{2},{3},{1,2},{1,3}}
%e A322698   {{1},{2},{3},{1,2},{1,3},{2,3}}
%t A322698 Table[Sum[SeriesCoefficient[Product[1+Times@@x/@s,{s,Union/@Select[Tuples[Range[n],2],OrderedQ]}],Sequence@@Table[{x[i],0,k},{i,n}]],{k,0,n-1}],{n,1,6}]
%o A322698 (PARI) for(n=1, 10, print1(A322698(n), ", ")) \\ See A295193 for script, _Andrew Howroyd_, Aug 28 2019
%Y A322698 Row sums of A333157.
%Y A322698 Cf. A058891, A059441, A116539, A283877, A295193, A319189, A319190, A319612, A319729.
%K A322698 nonn
%O A322698 0,2
%A A322698 _Gus Wiseman_, Dec 23 2018
%E A322698 a(10)-a(18) from _Andrew Howroyd_, Aug 28 2019