A362046 - OEIS

%I #12 Apr 18 2023 17:33:01

%S 0,0,1,1,3,3,9,8,25,23,75,68,235,213,759,695,2521,2325,8555,7941,

%T 29503,27561,103129,96861,364547,344003,1300819,1232566,4679471,

%U 4449849,16952161,16171117,61790441,59107889,226451035,217157068,833918839,801467551,3084255127

%N Number of nonempty subsets of {1..n} with mean n/2.

%F a(n) = (A070925(n) - 1)/2.

%F a(n) = A133406(n) - 1.

%F a(2n) = A212352(n) = A000980(n)/2 - 1.

%e The a(2) = 1 through a(7) = 8 subsets:

%e   {1}  {1,2}  {2}      {1,4}      {3}          {1,6}

%e               {1,3}    {2,3}      {1,5}        {2,5}

%e               {1,2,3}  {1,2,3,4}  {2,4}        {3,4}

%e                                   {1,2,6}      {1,2,4,7}

%e                                   {1,3,5}      {1,2,5,6}

%e                                   {2,3,4}      {1,3,4,6}

%e                                   {1,2,3,6}    {2,3,4,5}

%e                                   {1,2,4,5}    {1,2,3,4,5,6}

%e                                   {1,2,3,4,5}

%t Table[Length[Select[Subsets[Range[n]],Mean[#]==n/2&]],{n,0,15}]

%Y Using range 0..n gives A070925.

%Y Including the empty set gives A133406.

%Y Even bisection is A212352.

%Y For median instead of mean we have A361801, the doubling of A079309.

%Y A version for partitions is A361853, for median A361849.

%Y A000980 counts nonempty subsets of {1..2n-1} with mean n.

%Y A007318 counts subsets by length.

%Y A067538 counts partitions with integer mean, strict A102627.

%Y A231147 appears to count subsets by median, full-steps A013580.

%Y A327475 counts subsets with integer mean, A000975 integer median.

%Y A327481 counts subsets by integer mean.

%Y Cf. A006134, A024718, A057552, A349156, A359893, A361654, A361864.

%K nonn

%O 0,5

%A _Gus Wiseman_, Apr 12 2023