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%I #16 Jan 21 2023 17:58:25
%S 1,1,2,5,12,30,91,256,823,2656,9103,31876,116113,432824,1659692,
%T 6508521,26112327,106927561,446654187,1900858001,8236367607,
%U 36306790636,162724173883,741105774720,3428164417401,16099059101049,76722208278328,370903316203353,1818316254655097
%N Number of non-isomorphic strict connected multiset partitions of weight n.
%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.
%C Also the number of non-isomorphic connected T_0 multiset partitions of weight n. In a multiset partition, two vertices are equivalent if in every block the multiplicity of the first is equal to the multiplicity of the second. The T_0 condition means that there are no equivalent vertices.
%H Andrew Howroyd, <a href="/A319557/b319557.txt">Table of n, a(n) for n = 0..50</a>
%F Inverse Euler transform of A316980.
%e Non-isomorphic representatives of the a(4) = 12 strict connected multiset partitions:
%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 {{1},{1,1,1}}
%e {{1},{1,2,2}}
%e {{2},{1,2,2}}
%e {{3},{1,2,3}}
%e {{1,2},{2,2}}
%e {{1,3},{2,3}}
%e {{1},{2},{1,2}}
%e Non-isomorphic representatives of the a(4) = 12 connected T_0 multiset partitions:
%e {{1,1,1,1}}
%e {{1,2,2,2}}
%e {{1},{1,1,1}}
%e {{1},{1,2,2}}
%e {{2},{1,2,2}}
%e {{1,1},{1,1}}
%e {{1,2},{2,2}}
%e {{1,3},{2,3}}
%e {{1},{1},{1,1}}
%e {{1},{2},{1,2}}
%e {{2},{2},{1,2}}
%e {{1},{1},{1},{1}}
%Y Cf. A007716, A007718, A049311, A056156, A283877, A316980.
%Y Cf. A319558, A319559, A319560, A319564, A319565, A319566, A319567.
%K nonn
%O 0,3
%A _Gus Wiseman_, Sep 23 2018
%E Terms a(11) and beyond from _Andrew Howroyd_, Jan 19 2023
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