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%I #12 Jan 28 2024 18:10:42
%S 1,0,0,0,1,1,7,9,37,79,273,755,2648,8432,29872,104624,384759,1432655,
%T 5502563,21533141,86291313,352654980,1471073073,6253397866,
%U 27083003687,119399628021,535591458635,2443030798539,11326169401988,53343974825122,255121588496338
%N Number of non-isomorphic multiset partitions of weight n with no constant parts or vertices that appear in only one part.
%C Also the number of nonnegative integer matrices up to row and column permutations with sum of elements equal to n in which every row and column has at least two nonzero entries.
%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.
%H Andrew Howroyd, <a href="/A321760/b321760.txt">Table of n, a(n) for n = 0..50</a>
%e Non-isomorphic representatives of the a(4) = 1 through a(7) = 9 multiset partitions:
%e {{1,2},{1,2}} {{1,2},{1,2,2}} {{1,1,2},{1,2,2}} {{1,1,2},{1,2,2,2}}
%e {{1,2},{1,1,2,2}} {{1,2},{1,1,2,2,2}}
%e {{1,2},{1,2,2,2}} {{1,2},{1,2,2,2,2}}
%e {{1,2,2},{1,2,2}} {{1,2,2},{1,1,2,2}}
%e {{1,2,3},{1,2,3}} {{1,2,2},{1,2,2,2}}
%e {{1,2},{1,2},{1,2}} {{1,2,3},{1,2,3,3}}
%e {{1,2},{1,3},{2,3}} {{1,2},{1,2},{1,2,2}}
%e {{1,2},{1,3},{2,3,3}}
%e {{1,3},{2,3},{1,2,3}}
%o (PARI) Vec(G(20,1)) \\ G defined in A369286. - _Andrew Howroyd_, Jan 28 2024
%Y Row sums of A369286.
%Y Cf. A001970, A007716, A050535, A055884, A120733, A317533, A320665, A320798, A320801, A320808, A321407.
%K nonn
%O 0,7
%A _Gus Wiseman_, Nov 29 2018
%E a(11) onwards from _Andrew Howroyd_, Jan 27 2024
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