%I #9 Jan 05 2020 12:04:06
%S 1,1,2,4,20,140,1411
%N Number of non-isomorphic balanced reduced multisystems of weight n and maximum depth.
%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(2) = 2 through a(4) = 20 multisystems:
%e {1,1} {{1},{1,1}} {{{1}},{{1},{1,1}}}
%e {1,2} {{1},{1,2}} {{{1,1}},{{1},{1}}}
%e {{1},{2,3}} {{{1}},{{1},{1,2}}}
%e {{2},{1,1}} {{{1,1}},{{1},{2}}}
%e {{{1}},{{1},{2,2}}}
%e {{{1,1}},{{2},{2}}}
%e {{{1}},{{1},{2,3}}}
%e {{{1,1}},{{2},{3}}}
%e {{{1}},{{2},{1,1}}}
%e {{{1,2}},{{1},{1}}}
%e {{{1}},{{2},{1,2}}}
%e {{{1,2}},{{1},{2}}}
%e {{{1}},{{2},{1,3}}}
%e {{{1,2}},{{1},{3}}}
%e {{{1}},{{2},{3,4}}}
%e {{{1,2}},{{3},{4}}}
%e {{{2}},{{1},{1,1}}}
%e {{{2}},{{1},{1,3}}}
%e {{{2}},{{3},{1,1}}}
%e {{{2,3}},{{1},{1}}}
%Y The non-maximal version is A330474.
%Y Labeled versions are A330675 (strongly normal) and A330676 (normal).
%Y The case where the leaves are sets (as opposed to multisets) is A330677.
%Y The case with all atoms distinct is A000111.
%Y The case with all atoms equal is (also) A000111.
%Y Cf. A000311, A004114, A005121, A006472, A007716, A048816, A141268, A306186, A330470, A330655, A330664.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, Dec 27 2019