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%I A330663 #9 Jan 05 2020 12:04:06
%S A330663 1,1,2,4,20,140,1411
%N A330663 Number of non-isomorphic balanced reduced multisystems of weight n and maximum depth.
%C A330663 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 A330663 Non-isomorphic representatives of the a(2) = 2 through a(4) = 20 multisystems:
%e A330663 {1,1} {{1},{1,1}} {{{1}},{{1},{1,1}}}
%e A330663 {1,2} {{1},{1,2}} {{{1,1}},{{1},{1}}}
%e A330663 {{1},{2,3}} {{{1}},{{1},{1,2}}}
%e A330663 {{2},{1,1}} {{{1,1}},{{1},{2}}}
%e A330663 {{{1}},{{1},{2,2}}}
%e A330663 {{{1,1}},{{2},{2}}}
%e A330663 {{{1}},{{1},{2,3}}}
%e A330663 {{{1,1}},{{2},{3}}}
%e A330663 {{{1}},{{2},{1,1}}}
%e A330663 {{{1,2}},{{1},{1}}}
%e A330663 {{{1}},{{2},{1,2}}}
%e A330663 {{{1,2}},{{1},{2}}}
%e A330663 {{{1}},{{2},{1,3}}}
%e A330663 {{{1,2}},{{1},{3}}}
%e A330663 {{{1}},{{2},{3,4}}}
%e A330663 {{{1,2}},{{3},{4}}}
%e A330663 {{{2}},{{1},{1,1}}}
%e A330663 {{{2}},{{1},{1,3}}}
%e A330663 {{{2}},{{3},{1,1}}}
%e A330663 {{{2,3}},{{1},{1}}}
%Y A330663 The non-maximal version is A330474.
%Y A330663 Labeled versions are A330675 (strongly normal) and A330676 (normal).
%Y A330663 The case where the leaves are sets (as opposed to multisets) is A330677.
%Y A330663 The case with all atoms distinct is A000111.
%Y A330663 The case with all atoms equal is (also) A000111.
%Y A330663 Cf. A000311, A004114, A005121, A006472, A007716, A048816, A141268, A306186, A330470, A330655, A330664.
%K A330663 nonn,more
%O A330663 0,3
%A A330663 _Gus Wiseman_, Dec 27 2019