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%I #4 Nov 13 2018 12:54:25
%S 1,0,1,1,2,4,8,14,27,53,105
%N Number of non-isomorphic strict self-dual multiset partitions of weight n with no singletons.
%C Also the number of nonnegative integer symmetric matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, in which the rows are all different and none sums to 1.
%C The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.
%C The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.
%e Non-isomorphic representatives of the a(2) = 1 through a(7) = 14 multiset partitions:
%e {{11}} {{111}} {{1111}} {{11111}} {{111111}} {{1111111}}
%e {{11}{22}} {{11}{122}} {{111}{222}} {{111}{1222}}
%e {{11}{222}} {{112}{122}} {{111}{2222}}
%e {{12}{122}} {{11}{2222}} {{112}{1222}}
%e {{12}{1222}} {{11}{22222}}
%e {{22}{1122}} {{12}{12222}}
%e {{11}{22}{33}} {{122}{1122}}
%e {{12}{13}{23}} {{22}{11222}}
%e {{11}{12}{233}}
%e {{11}{22}{233}}
%e {{11}{22}{333}}
%e {{11}{23}{233}}
%e {{12}{13}{233}}
%e {{13}{23}{123}}
%Y Cf. A000219, A007716, A059201, A302545, A316980, A316983, A319560, A319616.
%Y Cf. A320796, A320797, A320798, A320804, A320811, A320812, A321401, A321404, A321405, A321406, A321407.
%K nonn,more
%O 0,5
%A _Gus Wiseman_, Nov 09 2018
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