This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A368598 #17 Feb 24 2024 11:03:25
%S A368598 1,1,2,6,17,52,173,585,2064,7520,28265,109501,437394,1799843,7629463,
%T A368598 33302834,149633151,691702799,3287804961,16058229900,80533510224,
%U A368598 414384339438,2185878202630,11811050484851,65318772618624,369428031895444,2135166786135671,12601624505404858
%N A368598 Number of non-isomorphic n-element sets of singletons or pairs of elements of {1..n}, or unlabeled loop-graphs with n edges and up to n vertices.
%C A368598 It doesn't matter for this sequence whether we use loops such as {x,x} or half-loops such as {x}.
%H A368598 Andrew Howroyd, <a href="/A368598/b368598.txt">Table of n, a(n) for n = 0..50</a>
%H A368598 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphLoop.html">Graph Loop</a>.
%F A368598 a(n) = A070166(n, n). - _Andrew Howroyd_, Jan 09 2024
%e A368598 Non-isomorphic representatives of the a(0) = 1 through a(4) = 17 set-systems:
%e A368598 {} {{1}} {{1},{2}} {{1},{2},{3}} {{1},{2},{3},{4}}
%e A368598 {{1},{1,2}} {{1},{2},{1,2}} {{1},{2},{3},{1,2}}
%e A368598 {{1},{2},{1,3}} {{1},{2},{3},{1,4}}
%e A368598 {{1},{1,2},{1,3}} {{1},{2},{1,2},{1,3}}
%e A368598 {{1},{1,2},{2,3}} {{1},{2},{1,2},{3,4}}
%e A368598 {{1,2},{1,3},{2,3}} {{1},{2},{1,3},{1,4}}
%e A368598 {{1},{2},{1,3},{2,3}}
%e A368598 {{1},{2},{1,3},{2,4}}
%e A368598 {{1},{3},{1,2},{2,4}}
%e A368598 {{1},{1,2},{1,3},{1,4}}
%e A368598 {{1},{1,2},{1,3},{2,3}}
%e A368598 {{1},{1,2},{1,3},{2,4}}
%e A368598 {{1},{1,2},{2,3},{3,4}}
%e A368598 {{2},{1,2},{1,3},{1,4}}
%e A368598 {{4},{1,2},{1,3},{2,3}}
%e A368598 {{1,2},{1,3},{1,4},{2,3}}
%e A368598 {{1,2},{1,3},{2,4},{3,4}}
%t A368598 brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{(Union@@m)[[i]],p[[i]]}, {i,Length[p]}])], {p,Permutations[Range[Length[Union@@m]]]}]]];
%t A368598 Table[Length[Union[brute /@ Subsets[Subsets[Range[n],{1,2}],{n}]]],{n,0,5}]
%o A368598 (PARI) a(n) = polcoef(G(n, O(x*x^n)), n) \\ G defined in A070166. - _Andrew Howroyd_, Jan 09 2024
%Y A368598 For any number of edges of any size we have A000612, covering A055621.
%Y A368598 For any number of edges we have A000666, A054921, A322700.
%Y A368598 The labeled version is A014068.
%Y A368598 Counting by weight gives A320663, or A339888 with loops {x,x}.
%Y A368598 The covering case is A368599.
%Y A368598 For edges of any size we have A368731, covering A368186.
%Y A368598 Row sums of A368836.
%Y A368598 A000085 counts set partitions into singletons or pairs.
%Y A368598 A001515 counts length-n set partitions into singletons or pairs.
%Y A368598 A100861 counts set partitions into singletons or pairs by number of pairs.
%Y A368598 A111924 counts set partitions into singletons or pairs by length.
%Y A368598 Cf. A001434, A007716, A007717, A058891, A070166, A122848, A124059, A283877, A302545, A322661, A339741, A339887, A370168.
%K A368598 nonn
%O A368598 0,3
%A A368598 _Gus Wiseman_, Jan 05 2024
%E A368598 Terms a(7) and beyond from _Andrew Howroyd_, Jan 09 2024