A366320 Irregular triangle read by rows where T(n,k) is the number of subsets of {1..n} without a subset summing to k.

1, 2, 2, 3, 4, 4, 3, 6, 6, 7, 8, 8, 6, 6, 9, 11, 11, 14, 14, 15, 16, 16, 12, 12, 9, 17, 17, 20, 20, 24, 27, 27, 30, 30, 31, 32, 32, 24, 24, 18, 17, 26, 31, 29, 35, 36, 43, 47, 50, 51, 56, 59, 59, 62, 62, 63

Offset: 1

Author

Gus Wiseman, Oct 12 2023

Keywords

• nonn
• tabf

Examples

			Triangle begins:
   1
   2  2  3
   4  4  3  6  6  7
   8  8  6  6  9 11 11 14 14 15
  16 16 12 12  9 17 17 20 20 24 27 27 30 30 31
  32 32 24 24 18 17 26 31 29 35 36 43 47 50 51 56 59 59 62 62 63
Row n = 3 counts the following subsets:
  {}     {}     {}   {}     {}     {}
  {2}    {1}    {1}  {1}    {1}    {1}
  {3}    {3}    {2}  {2}    {2}    {2}
  {2,3}  {1,3}       {3}    {3}    {3}
                     {1,2}  {1,2}  {1,2}
                     {2,3}  {1,3}  {1,3}
                                   {2,3}
		

Crossrefs

Row lengths are A000217.

The diagonal T(n,n) is A365377, complement A365376.

The complement is counted by A365381.

A000009 counts subsets summing to n.

A000124 counts distinct possible sums of subsets of {1..n}.

A124506 counts combination-free subsets, differences of A326083.

A365046 counts combination-full subsets, differences of A364914.

Cf. A007865, A085489, A093971, A103580, A151897, A326080, A364534, A365073, A365380.

Programs

Mathematica

Table[Length[Select[Subsets[Range[n]],FreeQ[Total/@Subsets[#],k]&]],{n,8},{k,n*(n+1)/2}]