A370818
Number of sets of nonempty subsets of {1..n} with only one possible way to choose a set of different vertices of each edge.
Original entry on oeis.org
1, 2, 6, 45, 1352, 157647, 63380093, 85147722812, 385321270991130
Offset: 0
The set-system {{2},{1,2},{2,4},{1,3,4}} has unique choice (2,1,4,3) so is counted under a(4).
Factorizations of this type are counted by
A370645.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
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Table[Length[Select[Subsets[Subsets[Range[n]]], Length[Union[Sort/@Select[Tuples[#],UnsameQ@@#&]]]==1&]],{n,0,3}]
A138237
Number of unlabeled graphs with at least one cycle in which every connected component has at most one cycle.
Original entry on oeis.org
1, 3, 9, 26, 71, 197, 543, 1507, 4186, 11722, 32883, 92724, 262179, 743792, 2115019, 6028779, 17217093, 49258009, 141142096, 404997704, 1163569094, 3346830818, 9636723582, 27774427243, 80121104084, 231317022483, 668346261557
Offset: 3
a(9)=543 since we have several cases, with one unicyclic graph, or two, or three. Namely,
-One triangle and a forest of order 6, or 20 graphs.
-One unicyclic graph with 4 nodes and a forest of order 5, or 20 graphs.
-One unicyclic graph with 5 nodes and a forest of order 4, or 30 graphs.
-One unicyclic graph with 6 nodes and a forest of order 3, or 39 graphs.
-One unicyclic graph of 7 nodes and a forest of order 2, or 66 graphs.
-One unicyclic graph of 8 nodes and an isolated vertex, or 89 graphs.
-One unicyclic graph of 9 nodes, or 240 graphs.
-Two triangles and a forest of order 3, or 3 graphs.
-One triangle plus one unicyclic graph of 4 nodes plus a forest of order 2, or 4 graphs.
-One triangle plus one unicyclic graph of 5 nodes plus an isolated vertex, or 5 graphs.
-One triangle plus one unicyclic graph of 6 nodes, or 13 graphs.
-Two unicyclic graphs of 4 nodes and an isolated vertex, or C(2+2-1,2)=3 graphs.
-One unicyclic graph of 5 nodes and another of 4 nodes, or 10 graphs.
-Three triangles, or 1 graph.
Total = 543.