A365541 Irregular triangle read by rows where T(n,k) is the number of subsets of {1..n} containing two distinct elements summing to k = 3..2n-1.

1, 2, 2, 2, 4, 4, 7, 4, 4, 8, 8, 14, 14, 14, 8, 8, 16, 16, 28, 28, 37, 28, 28, 16, 16, 32, 32, 56, 56, 74, 74, 74, 56, 56, 32, 32, 64, 64, 112, 112, 148, 148, 175, 148, 148, 112, 112, 64, 64, 128, 128, 224, 224, 296, 296, 350, 350, 350, 296, 296, 224, 224, 128, 128

Offset: 2

Author

Gus Wiseman, Sep 15 2023

Keywords

• nonn
• tabf

Comments

Rows are palindromic.

Examples

			Triangle begins:
    1
    2    2    2
    4    4    7    4    4
    8    8   14   14   14    8    8
   16   16   28   28   37   28   28   16   16
   32   32   56   56   74   74   74   56   56   32   32
Row n = 4 counts the following subsets:
  {1,2}      {1,3}      {1,4}      {2,4}      {3,4}
  {1,2,3}    {1,2,3}    {2,3}      {1,2,4}    {1,3,4}
  {1,2,4}    {1,3,4}    {1,2,3}    {2,3,4}    {2,3,4}
  {1,2,3,4}  {1,2,3,4}  {1,2,4}    {1,2,3,4}  {1,2,3,4}
                        {1,3,4}
                        {2,3,4}
                        {1,2,3,4}
		

Crossrefs

Row lengths are A005408.

The case counting only length-2 subsets is A008967.

Column k = n + 1 appears to be A167762.

The version for all subsets (instead of just pairs) is A365381.

Column k = n is A365544.

A000009 counts subsets summing to n.

A007865/A085489/A151897 count certain types of sum-free subsets.

A046663 counts partitions with no submultiset summing to k, strict A365663.

A093971/A088809/A364534 count certain types of sum-full subsets.

A365543 counts partitions with a submultiset summing to k, strict A365661.

Cf. A068911, A095944, A238628, A288728, A326083, A364272, A365376, A365377.

Programs

Mathematica

Table[Length[Select[Subsets[Range[n]], MemberQ[Total/@Subsets[#,{2}],k]&]], {n,2,11}, {k,3,2n-1}]