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%I #15 Jul 20 2018 11:25:24
%S 1,2,5,28,277,3985,76117,1833187,53756682,1871041538,75809298105,
%T 3521419837339,185235838688677,10923147890901151,715989783027216302,
%U 51793686238309903860,4109310551278549543317,355667047514571431358297,33422937748872646130124797
%N Number of connected multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n}.
%C Note that all connected multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n} are strict except for (123...n)(123...n).
%F Logarithmic transform of A020555.
%e The a(2) = 5 connected multiset partitions of {1, 1, 2, 2} are (1122), (1)(122), (2)(112), (12)(12), (1)(2)(12). The multiset partitions (11)(22), (1)(1)(22), (2)(2)(11), (1)(1)(2)(2) are not connected.
%t nn=10;
%t ser=Exp[-3/2+Exp[x]/2]*Sum[Exp[Binomial[n+1,2]*x]/n!,{n,0,3*nn}];
%t Round/@(CoefficientList[Series[1+Log[ser],{x,0,nn}],x]*Array[Factorial,nn+1,0]) (* based on _Jean-François Alcover_ after _Vladeta Jovovic_ *)
%t (*second program *)
%t csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t Length/@Table[Select[mps[Ceiling[Range[1/2,n,1/2]]],Length[csm[#]]==1&],{n,4}]
%Y Cf. A001055, A007716, A007717, A007719, A020555, A045778, A061742, A094574, A162247, A316974.
%K nonn
%O 0,2
%A _Gus Wiseman_, Jul 17 2018
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