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A330679 Number of balanced reduced multisystems whose atoms constitute an integer partition of n.

%I A330679 #8 Jan 20 2024 16:07:16
%S A330679 1,1,2,4,12,40,180,936,5820,41288,331748,2968688,29307780,316273976,
%T A330679 3704154568,46788812168,634037127612,9174782661984,141197140912208,
%U A330679 2302765704401360,39671953757409256,719926077632193848,13726066030661998220,274313334040504957368
%N A330679 Number of balanced reduced multisystems whose atoms constitute an integer partition of n.
%C A330679 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.
%H A330679 Andrew Howroyd, <a href="/A330679/b330679.txt">Table of n, a(n) for n = 0..200</a>
%F A330679 a(n > 1) = 2 * A318813(n).
%e A330679 The a(0) = 1 through a(4) = 12 multisystems:
%e A330679   {}  {1}  {2}    {3}          {4}
%e A330679            {1,1}  {1,2}        {1,3}
%e A330679                   {1,1,1}      {2,2}
%e A330679                   {{1},{1,1}}  {1,1,2}
%e A330679                                {1,1,1,1}
%e A330679                                {{1},{1,2}}
%e A330679                                {{2},{1,1}}
%e A330679                                {{1},{1,1,1}}
%e A330679                                {{1,1},{1,1}}
%e A330679                                {{1},{1},{1,1}}
%e A330679                                {{{1}},{{1},{1,1}}}
%e A330679                                {{{1,1}},{{1},{1}}}
%t A330679 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A330679 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A330679 totm[m_]:=Prepend[Join@@Table[totm[p],{p,Select[mps[m],1<Length[#]<Length[m]&]}],m];
%t A330679 Table[Sum[Length[totm[m]],{m,IntegerPartitions[n]}],{n,0,5}]
%Y A330679 The case where the atoms are all 1's is A318813 = a(n)/2.
%Y A330679 The version where the atoms constitute a strongly normal multiset is A330475.
%Y A330679 The version where the atoms cover an initial interval is A330655.
%Y A330679 The maximum-depth version is A330726.
%Y A330679 Cf. A000041, A000111, A000669, A001970, A002846, A005121, A141268, A196545, A213427, A318812, A320160, A330474.
%K A330679 nonn
%O A330679 0,3
%A A330679 _Gus Wiseman_, Dec 31 2019
%E A330679 a(12) onwards from _Andrew Howroyd_, Jan 20 2024