%I #6 Apr 27 2020 09:43:15
%S 1,1,1,2,11,81,859
%N Number of non-isomorphic balanced reduced multisystems of weight n and maximum depth whose leaves (which are multisets of atoms) are sets.
%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(0) = 1 through a(4) = 11 multisystems:
%e {} {1} {1,2} {{1},{1,2}} {{{1}},{{1},{1,2}}}
%e {{1},{2,3}} {{{1}},{{1},{2,3}}}
%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},{1,3}}}
%e {{{2,3}},{{1},{1}}}
%Y The version with all distinct atoms is A000111.
%Y Non-isomorphic set multipartitions are A049311.
%Y The (non-maximal) tree version is A330626.
%Y Allowing leaves to be multisets gives A330663.
%Y The case with prescribed degrees is A330664.
%Y The version allowing all depths is A330668.
%Y Cf. A000669, A001678, A004114, A005121, A007716, A141268, A283877, A306186, A330465, A330470, A330624.
%K nonn,more
%O 0,4
%A _Gus Wiseman_, Dec 30 2019