%I #4 Nov 16 2018 07:48:59
%S 1,1,1,3,6,16,27,71,135,309,621
%N Number of non-isomorphic self-dual multiset partitions of weight n whose part sizes are relatively prime.
%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, with relatively prime row sums (or column sums).
%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(1) = 1 through a(5) = 16 multiset partitions:
%e {{1}} {{1}{2}} {{1}{22}} {{1}{222}} {{11}{122}}
%e {{2}{12}} {{2}{122}} {{11}{222}}
%e {{1}{2}{3}} {{1}{1}{23}} {{12}{122}}
%e {{1}{2}{33}} {{1}{2222}}
%e {{1}{3}{23}} {{2}{1222}}
%e {{1}{2}{3}{4}} {{1}{22}{33}}
%e {{1}{23}{23}}
%e {{1}{2}{333}}
%e {{1}{3}{233}}
%e {{2}{12}{33}}
%e {{2}{13}{23}}
%e {{3}{3}{123}}
%e {{1}{2}{2}{34}}
%e {{1}{2}{3}{44}}
%e {{1}{2}{4}{34}}
%e {{1}{2}{3}{4}{5}}
%Y Cf. A000219, A007716, A120733, A138178, A316983, A319616.
%Y Cf. A320796, A320797, A320800, A320805, A320806, A320809, A320811, A320813, A321283, A321410, A321411, A321413.
%K nonn,more
%O 0,4
%A _Gus Wiseman_, Nov 16 2018