%I #10 Dec 30 2020 17:12:00
%S 1,6,27,169,1029,7817,61006,547537,5202009,54506262,606311524,
%T 7299051826,92985064466,1264720212352,18137495642192,275078184766323,
%U 4379514178076452,73235806332442156,1280229713195027792,23381809052104639236,444740694108284116235,8801030741502964613534
%N Number of multiset partitions of multiset partitions of strongly normal multisets of size n.
%C A multiset is normal if it spans an initial interval of positive integers, and strongly normal if in addition it has weakly decreasing multiplicities.
%e The a(2) = 6 multiset partitions of multiset partitions:
%e {{{1,1}}}
%e {{{1,2}}}
%e {{{1},{1}}}
%e {{{1},{2}}}
%e {{{1}},{{1}}}
%e {{{1}},{{2}}}
%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
%t Table[Sum[Length[mps[m]],{m,Join@@mps/@strnorm[n]}],{n,6}]
%o (PARI) \\ See links in A339645 for combinatorial species functions.
%o seq(n)={my(A=symGroupSeries(n)); StronglyNormalLabelingsSeq(sExp(sExp(A))-1)} \\ _Andrew Howroyd_, Dec 30 2020
%Y Cf. A001970, A007716, A050336, A255906, A269134, A317533, A317791, A318564, A318566.
%K nonn
%O 1,2
%A _Gus Wiseman_, Aug 29 2018
%E Terms a(9) and beyond from _Andrew Howroyd_, Dec 30 2020