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%I #4 Oct 29 2018 09:55:50
%S 1,1,4,10,31,87,272,835,2673,8805,29583
%N Number of non-isomorphic knapsack multiset partitions of weight n.
%C A multiset partition is knapsack if every distinct submultiset of the parts has a different multiset union.
%e Non-isomorphic representatives of the a(1) = 1 through a(4) = 31 knapsack multiset partitions:
%e {{1}} {{1,1}} {{1,1,1}} {{1,1,1,1}}
%e {{1,2}} {{1,2,2}} {{1,1,2,2}}
%e {{1},{1}} {{1,2,3}} {{1,2,2,2}}
%e {{1},{2}} {{1},{1,1}} {{1,2,3,3}}
%e {{1},{2,2}} {{1,2,3,4}}
%e {{1},{2,3}} {{1},{1,1,1}}
%e {{2},{1,2}} {{1,1},{1,1}}
%e {{1},{1},{1}} {{1},{1,2,2}}
%e {{1},{2},{2}} {{1,1},{2,2}}
%e {{1},{2},{3}} {{1,2},{1,2}}
%e {{1},{2,2,2}}
%e {{1,2},{2,2}}
%e {{1},{2,3,3}}
%e {{1,2},{3,3}}
%e {{1},{2,3,4}}
%e {{1,2},{3,4}}
%e {{1,3},{2,3}}
%e {{2},{1,2,2}}
%e {{3},{1,2,3}}
%e {{1},{1},{2,2}}
%e {{1},{1},{2,3}}
%e {{1},{2},{2,2}}
%e {{1},{2},{3,3}}
%e {{1},{2},{3,4}}
%e {{1},{3},{2,3}}
%e {{2},{2},{1,2}}
%e {{1},{1},{1},{1}}
%e {{1},{1},{2},{2}}
%e {{1},{2},{2},{2}}
%e {{1},{2},{3},{3}}
%e {{1},{2},{3},{4}}
%e Missing from this list are {{1},{1},{1,1}} and {{1},{2},{1,2}}, which are not knapsack.
%Y Cf. A002219, A006827, A007716, A108917, A275972, A276024, A292886, A316983, A319616.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, Oct 28 2018
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