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%I #8 Apr 27 2020 09:44:24
%S 1,1,2,7,48,424
%N Number of non-isomorphic balanced reduced multisystems of weight 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. The weight of an atom is 1, while the weight of a multiset is the sum of weights of its elements.
%e Non-isomorphic representatives of the a(3) = 7 multisystems:
%e {1,1,1}
%e {1,1,2}
%e {1,2,3}
%e {{1},{1,1}}
%e {{1},{1,2}}
%e {{1},{2,3}}
%e {{2},{1,1}}
%e Non-isomorphic representatives of the a(4) = 48 multisystems:
%e {1,1,1,1} {{1},{1,1,1}} {{{1}},{{1},{1,1}}}
%e {1,1,1,2} {{1,1},{1,1}} {{{1,1}},{{1},{1}}}
%e {1,1,2,2} {{1},{1,1,2}} {{{1}},{{1},{1,2}}}
%e {1,1,2,3} {{1,1},{1,2}} {{{1,1}},{{1},{2}}}
%e {1,2,3,4} {{1},{1,2,2}} {{{1}},{{1},{2,2}}}
%e {{1,1},{2,2}} {{{1,1}},{{2},{2}}}
%e {{1},{1,2,3}} {{{1}},{{1},{2,3}}}
%e {{1,1},{2,3}} {{{1,1}},{{2},{3}}}
%e {{1,2},{1,2}} {{{1}},{{2},{1,1}}}
%e {{1,2},{1,3}} {{{1,2}},{{1},{1}}}
%e {{1},{2,3,4}} {{{1}},{{2},{1,2}}}
%e {{1,2},{3,4}} {{{1,2}},{{1},{2}}}
%e {{2},{1,1,1}} {{{1}},{{2},{1,3}}}
%e {{2},{1,1,3}} {{{1,2}},{{1},{3}}}
%e {{1},{1},{1,1}} {{{1}},{{2},{3,4}}}
%e {{1},{1},{1,2}} {{{1,2}},{{3},{4}}}
%e {{1},{1},{2,2}} {{{2}},{{1},{1,1}}}
%e {{1},{1},{2,3}} {{{2}},{{1},{1,3}}}
%e {{1},{2},{1,1}} {{{2}},{{3},{1,1}}}
%e {{1},{2},{1,2}} {{{2,3}},{{1},{1}}}
%e {{1},{2},{1,3}}
%e {{1},{2},{3,4}}
%e {{2},{3},{1,1}}
%Y Labeled versions are A330475 (strongly normal) and A330655 (normal).
%Y The case where the atoms are all different is A318813.
%Y The case where the atoms are all equal is (also) A318813.
%Y The labeled case of set partitions is A005121.
%Y The labeled case of integer partitions is A330679.
%Y The case of maximal depth is A330663.
%Y The version where leaves are sets (as opposed to multisets) is A330668.
%Y Cf. A000311, A000669, A001678, A002846, A004114, A007716, A048816, A213427, A306186, A320154, A320160, A330470, A330666.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, Dec 26 2019
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