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%I #5 Jan 04 2020 09:52:40
%S 1,1,2,3,7,17,54,199,869,4341,24514,154187
%N Number of balanced reduced multisystems of maximum depth whose atoms are positive integers summing to n.
%C A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem.
%e The a(1) = 1 through a(5) = 17 multisystems (commas elided):
%e {1} {2} {3} {4} {5}
%e {11} {12} {22} {23}
%e {{1}{11}} {13} {14}
%e {{1}{12}} {{1}{13}}
%e {{2}{11}} {{1}{22}}
%e {{{1}}{{1}{11}}} {{2}{12}}
%e {{{11}}{{1}{1}}} {{3}{11}}
%e {{{1}}{{1}{12}}}
%e {{{11}}{{1}{2}}}
%e {{{1}}{{2}{11}}}
%e {{{12}}{{1}{1}}}
%e {{{2}}{{1}{11}}}
%e {{{{1}}}{{{1}}{{1}{11}}}}
%e {{{{1}}}{{{11}}{{1}{1}}}}
%e {{{{1}{1}}}{{{1}}{{11}}}}
%e {{{{1}{11}}}{{{1}}{{1}}}}
%e {{{{11}}}{{{1}}{{1}{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 totm[m_]:=Prepend[Join@@Table[totm[p],{p,Select[mps[m],1<Length[#]<Length[m]&]}],m];
%t Table[Sum[Length[Select[totm[m],Depth[#]==If[Length[m]<=1,2,Length[m]]&]],{m,IntegerPartitions[n]}],{n,0,5}]
%Y The case with all atoms equal to 1 is A000111.
%Y The non-maximal version is A330679.
%Y A tree version is A320160.
%Y Cf. A000669, A002846, A005121, A141268, A196545, A213427, A317145, A318813, A330663, A330665, A330675, A330676, A330728.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, Jan 03 2020
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