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 A368726 #15 Jan 11 2024 20:04:14
%S A368726 1,1,3,3,8,10,26,38,93,161,381,732,1721,3566,8369,18316,43280,98401,
%T A368726 234959,549628,1327726,3175670,7763500,18905703,46762513,115613599,
%U A368726 289185492,724438500,1831398264,4641907993,11853385002,30365353560
%N A368726 Number of non-isomorphic connected multiset partitions of weight n into singletons or pairs.
%H A368726 Andrew Howroyd, <a href="/A368726/b368726.txt">Table of n, a(n) for n = 0..50</a>
%F A368726 Inverse Euler transform of A320663.
%e A368726 Non-isomorphic representatives of the a(1) = 1 through a(5) = 10 multiset partitions:
%e A368726 {{1}} {{1,1}} {{1},{1,1}} {{1,1},{1,1}} {{1},{1,1},{1,1}}
%e A368726 {{1,2}} {{2},{1,2}} {{1,2},{1,2}} {{1},{1,2},{2,2}}
%e A368726 {{1},{1}} {{1},{1},{1}} {{1,2},{2,2}} {{2},{1,2},{1,2}}
%e A368726 {{1,3},{2,3}} {{2},{1,2},{2,2}}
%e A368726 {{1},{1},{1,1}} {{2},{1,3},{2,3}}
%e A368726 {{1},{2},{1,2}} {{3},{1,3},{2,3}}
%e A368726 {{2},{2},{1,2}} {{1},{1},{1},{1,1}}
%e A368726 {{1},{1},{1},{1}} {{1},{2},{2},{1,2}}
%e A368726 {{2},{2},{2},{1,2}}
%e A368726 {{1},{1},{1},{1},{1}}
%t A368726 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A368726 mpm[n_]:=Join@@Table[Union[Sort[Sort/@(#/.x_Integer:>s[[x]])]& /@ sps[Range[n]]],{s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}];
%t A368726 csm[s_]:=With[{c=Select[Subsets[Range[Length[s]], {2}],Length[Intersection@@s[[#]]]>0&]}, If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
%t A368726 brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}]]];
%t A368726 Table[Length[Union[brute /@ Select[mpm[n], Max@@Length/@#<=2&&Length[csm[#]]<=1&]]],{n,0,8}]
%Y A368726 For edges of any size we have A007718.
%Y A368726 This is the connected case of A320663.
%Y A368726 The case of singletons and strict pairs is A368727, Euler transform A339888.
%Y A368726 A000085, A100861, A111924 count set partitions into singletons or pairs.
%Y A368726 A007716 counts non-isomorphic multiset partitions, into pairs A007717.
%Y A368726 A062740 counts connected loop-graphs, unlabeled A054921.
%Y A368726 A320732 counts factorizations into primes or semiprimes, strict A339839.
%Y A368726 A322661 counts covering loop-graphs, unlabeled A322700.
%Y A368726 Cf. A001515, A000666, A122848, A283877, A302545, A320462, A321405, A368186, A368598, A368599, A368731.
%K A368726 nonn
%O A368726 0,3
%A A368726 _Gus Wiseman_, Jan 06 2024