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%I #8 Jan 01 2021 14:22:04
%S 1,6,36,274,2408,24440,279172,3542798,49354816,747851112,12231881948,
%T 214593346534,4016624367288,79843503990710,1678916979373760,
%U 37215518578700028,866953456654946948,21167221410812128266,540346299720320080828,14390314687100383124540,399023209689817997883900
%N Number of multiset partitions of multiset partitions of normal multisets of size n.
%C A multiset is normal if it spans an initial interval of positive integers.
%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 allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
%t Table[Sum[Length[mps[m]],{m,Join@@mps/@allnorm[n]}],{n,6}]
%o (PARI) \\ See links in A339645 for combinatorial species functions.
%o seq(n)={my(A=symGroupSeries(n)); NormalLabelingsSeq(sExp(sExp(A))-1)} \\ _Andrew Howroyd_, Jan 01 2021
%Y Cf. A001970, A007716, A050336, A255906, A269134, A317533, A317791, A318566.
%K nonn
%O 1,2
%A _Gus Wiseman_, Aug 29 2018
%E Terms a(8) and beyond from _Andrew Howroyd_, Jan 01 2021
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