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%I #6 Apr 05 2023 08:29:49
%S 1,0,3,2,12,18,101,232,1547,3768,24974,116728
%N Number of set partitions of {1..n} such that the mean of the means of the blocks is an integer.
%e The set partition y = {{1,4},{2,5},{3}} has block-means {5/2,7/2,3}, with mean 3, so y is counted under a(5).
%e The a(1) = 1 through a(5) = 12 set partitions:
%e {{1}} . {{123}} {{1}{234}} {{12345}}
%e {{13}{2}} {{123}{4}} {{1245}{3}}
%e {{1}{2}{3}} {{135}{24}}
%e {{15}{234}}
%e {{1}{234}{5}}
%e {{12}{3}{45}}
%e {{135}{2}{4}}
%e {{14}{25}{3}}
%e {{15}{24}{3}}
%e {{1}{24}{3}{5}}
%e {{15}{2}{3}{4}}
%e {{1}{2}{3}{4}{5}}
%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t Table[Length[Select[sps[Range[n]],IntegerQ[Mean[Mean/@#]]&]],{n,6}]
%Y For median instead of mean we have A361864.
%Y For sum instead of outer mean we have A361866, median A361911.
%Y A000110 counts set partitions.
%Y A067538 counts partitions with integer mean, ranks A326836, strict A102627.
%Y A308037 appears to count set partitions whose block-sizes have integer mean.
%Y A327475 counts subsets with integer mean, median A000975.
%Y Cf. A007837, A035470, A038041, A275714, A275780, A326512, A326513, A326521, A326537, A327481.
%K nonn,more
%O 1,3
%A _Gus Wiseman_, Apr 04 2023
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