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%I #17 Feb 03 2022 16:45:59
%S 1,1,4,4,12,4,48,4,183,297,1186,4,33950,4,139527,1529608,4726356,4,
%T 229255536,4,3705777010,36279746314,13764663019,4,14096735197959,
%U 5194673049514,7907992957755,2977586461058927,13426396910491001,4,1350012288268171854,4,59487352224070807287
%N Number of uniform multiset partitions of uniform multisets of size n whose union is an initial interval of positive integers.
%C A multiset is uniform if all multiplicities are equal. A multiset partition is uniform if all parts have the same size.
%H Andrew Howroyd, <a href="/A322785/b322785.txt">Table of n, a(n) for n = 0..100</a>
%F a(n) = 4 <=> n in { A000040 }. - _Alois P. Heinz_, Feb 03 2022
%e The a(1) = 1 though a(6) = 48 multiset partitions:
%e {1} {11} {111} {1111} {11111} {111111}
%e {12} {123} {1122} {12345} {111222}
%e {1}{1} {1}{1}{1} {1234} {1}{1}{1}{1}{1} {112233}
%e {1}{2} {1}{2}{3} {11}{11} {1}{2}{3}{4}{5} {123456}
%e {11}{22} {111}{111}
%e {12}{12} {111}{222}
%e {12}{34} {112}{122}
%e {13}{24} {112}{233}
%e {14}{23} {113}{223}
%e {1}{1}{1}{1} {122}{133}
%e {1}{1}{2}{2} {123}{123}
%e {1}{2}{3}{4} {123}{456}
%e {124}{356}
%e {125}{346}
%e {126}{345}
%e {134}{256}
%e {135}{246}
%e {136}{245}
%e {145}{236}
%e {146}{235}
%e {156}{234}
%e {11}{11}{11}
%e {11}{12}{22}
%e {11}{22}{33}
%e {11}{23}{23}
%e {12}{12}{12}
%e {12}{12}{33}
%e {12}{13}{23}
%e {12}{34}{56}
%e {12}{35}{46}
%e {12}{36}{45}
%e {13}{13}{22}
%e {13}{24}{56}
%e {13}{25}{46}
%e {13}{26}{45}
%e {14}{23}{56}
%e {14}{25}{36}
%e {14}{26}{35}
%e {15}{23}{46}
%e {15}{24}{36}
%e {15}{26}{34}
%e {16}{23}{45}
%e {16}{24}{35}
%e {16}{25}{34}
%e {1}{1}{1}{1}{1}{1}
%e {1}{1}{1}{2}{2}{2}
%e {1}{1}{2}{2}{3}{3}
%e {1}{2}{3}{4}{5}{6}
%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 Table[Sum[Length[Select[mps[m],SameQ@@Length/@#&]],{m,Table[Join@@Table[Range[n/d],{d}],{d,Divisors[n]}]}],{n,8}]
%Y Row sums of A322788.
%Y Cf. A000040, A038041, A072774, A100778, A299353, A306017, A306018, A306021, A317583, A317584, A319056, A319189, A321721, A322705, A322784, A322788.
%K nonn
%O 0,3
%A _Gus Wiseman_, Dec 26 2018
%E More terms from _Alois P. Heinz_, Jan 30 2019
%E Terms a(14) and beyond from _Andrew Howroyd_, Feb 03 2022
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