A361802 Irregular triangle read by rows where T(n,k) is the number of k-subsets of {-n+1,...,n} with sum 0, for k = 1,...,2n-1.

1, 1, 1, 1, 1, 2, 3, 2, 1, 1, 3, 6, 7, 5, 2, 1, 1, 4, 10, 16, 18, 14, 8, 3, 1, 1, 5, 15, 31, 46, 51, 43, 27, 12, 3, 1, 1, 6, 21, 53, 98, 139, 155, 134, 88, 43, 16, 4, 1, 1, 7, 28, 83, 184, 319, 441, 486, 424, 293, 161, 68, 21, 4, 1

Offset: 1

Author

Gus Wiseman, Apr 10 2023

Keywords

• nonn
• tabf

Comments

Also the number of k-subsets of {1,...,2n} with mean n.

Examples

			Triangle begins:
   1
   1   1   1
   1   2   3   2   1
   1   3   6   7   5   2   1
   1   4  10  16  18  14   8   3   1
   1   5  15  31  46  51  43  27  12   3   1
   1   6  21  53  98 139 155 134  88  43  16   4   1
   1   7  28  83 184 319 441 486 424 293 161  68  21   4   1
Row n = 3 counts the following subsets:
  {0}  {-1,1}  {-1,0,1}   {-2,-1,0,3}  {-2,-1,0,1,2}
       {-2,2}  {-2,0,2}   {-2,-1,1,2}
               {-2,-1,3}
		

Crossrefs

Row lengths are A005408.

Row sums are A212352.

A007318 counts subsets by length.

A067538 counts partitions with integer mean.

A231147 counts subsets by median.

A327475 counts subsets with integer mean, median A000975.

A327481 counts subsets by mean.

Cf. A006134, A013580, A024718, A079309, A326512, A349156, A361654, A362046.

Programs

Mathematica

Table[Length[Select[Subsets[Range[-n+1,n],{k}],Total[#]==0&]],{n,6},{k,2n-1}]