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%I #4 Jan 04 2020 09:52:55
%S 1,1,1,1,3,2,1,5,9,5,1,9,28,36,16,1,13,69,160,164,61,1,20,160,580,
%T 1022,855,272,1,28,337,1837,4996,7072,4988,1385
%N Triangle read by rows where T(n,k) is the number of balanced reduced multisystems of depth k with n equal atoms.
%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.
%F T(n,3) = A000041(n) - 2.
%F T(n,4) = A001970(n) - 3 * A000041(n) + 3.
%e Triangle begins:
%e 1
%e 1 1
%e 1 3 2
%e 1 5 9 5
%e 1 9 28 36 16
%e 1 13 69 160 164 61
%e 1 20 160 580 1022 855 272
%e 1 28 337 1837 4996 7072 4988 1385
%e Row n = 5 counts the following multisystems (strings of 1's are replaced by their lengths):
%e 5 {1,4} {{1},{1,3}} {{{1}},{{1},{1,2}}}
%e {2,3} {{1},{2,2}} {{{1,1}},{{1},{2}}}
%e {1,1,3} {{2},{1,2}} {{{1}},{{2},{1,1}}}
%e {1,2,2} {{3},{1,1}} {{{1,2}},{{1},{1}}}
%e {1,1,1,2} {{1},{1,1,2}} {{{2}},{{1},{1,1}}}
%e {{1,1},{1,2}}
%e {{2},{1,1,1}}
%e {{1},{1},{1,2}}
%e {{1},{2},{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[Length[Select[totm[ConstantArray[1,n]],Depth[#]==k&]],{n,2,6},{k,2,n}]
%Y Row sums are A318813.
%Y Column k = 3 is A007042.
%Y Column k = 4 is A001970(n) - 3*A000041(n) + 3.
%Y Column k = n is A000111.
%Y Row n is row prime(n) of A330727.
%Y Cf. A000669, A001055, A002846, A005121, A196545, A213427, A318812, A320160, A330474, A330475, A330655, A330667, A330679.
%K nonn,more,tabl
%O 2,5
%A _Gus Wiseman_, Jan 03 2020
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