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

1,6

Comments

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

Links

Table of n, a(n) for n=1..64.

Example

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}

Mathematica

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

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.

Sequence in context: A333271 A208233 A176270 * A086437 A027907 A026323

Adjacent sequences: A361799 A361800 A361801 * A361803 A361804 A361805

Keyword

nonn,tabf

Author

Gus Wiseman, Apr 10 2023

Status

approved