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 A330784 Triangle read by rows where T(n,k) is the number of balanced reduced multisystems of depth k with n equal atoms. 3
 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, 1022, 855, 272, 1, 28, 337, 1837, 4996, 7072, 4988, 1385 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 2,5 COMMENTS 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. LINKS FORMULA T(n,3) = A000041(n) - 2. T(n,4) = A001970(n) - 3 * A000041(n) + 3. EXAMPLE Triangle begins:     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 1022  855  272     1   28  337 1837 4996 7072 4988 1385 Row n = 5 counts the following multisystems (strings of 1's are replaced by their lengths):   5  {1,4}      {{1},{1,3}}      {{{1}},{{1},{1,2}}}      {2,3}      {{1},{2,2}}      {{{1,1}},{{1},{2}}}      {1,1,3}    {{2},{1,2}}      {{{1}},{{2},{1,1}}}      {1,2,2}    {{3},{1,1}}      {{{1,2}},{{1},{1}}}      {1,1,1,2}  {{1},{1,1,2}}    {{{2}},{{1},{1,1}}}                 {{1,1},{1,2}}                 {{2},{1,1,1}}                 {{1},{1},{1,2}}                 {{1},{2},{1,1}} MATHEMATICA sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}]; mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]]; totm[m_]:=Prepend[Join@@Table[totm[p], {p, Select[mps[m], 1

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Last modified July 27 15:29 EDT 2021. Contains 346307 sequences. (Running on oeis4.)