A362046 Number of nonempty subsets of {1..n} with mean n/2.

0, 0, 1, 1, 3, 3, 9, 8, 25, 23, 75, 68, 235, 213, 759, 695, 2521, 2325, 8555, 7941, 29503, 27561, 103129, 96861, 364547, 344003, 1300819, 1232566, 4679471, 4449849, 16952161, 16171117, 61790441, 59107889, 226451035, 217157068, 833918839, 801467551, 3084255127

Offset: 0

Author

Gus Wiseman, Apr 12 2023

Keywords

• nonn

Examples

			The a(2) = 1 through a(7) = 8 subsets:
  {1}  {1,2}  {2}      {1,4}      {3}          {1,6}
              {1,3}    {2,3}      {1,5}        {2,5}
              {1,2,3}  {1,2,3,4}  {2,4}        {3,4}
                                  {1,2,6}      {1,2,4,7}
                                  {1,3,5}      {1,2,5,6}
                                  {2,3,4}      {1,3,4,6}
                                  {1,2,3,6}    {2,3,4,5}
                                  {1,2,4,5}    {1,2,3,4,5,6}
                                  {1,2,3,4,5}
		

Crossrefs

Using range 0..n gives A070925.

Including the empty set gives A133406.

Even bisection is A212352.

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

A version for partitions is A361853, for median A361849.

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

A007318 counts subsets by length.

A067538 counts partitions with integer mean, strict A102627.

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

A327475 counts subsets with integer mean, A000975 integer median.

A327481 counts subsets by integer mean.

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

Programs

Mathematica

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

Formula

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

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

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