A357487 Number of integer partitions of n with the same length as reverse-alternating sum.

1, 1, 0, 0, 0, 1, 0, 2, 0, 4, 0, 5, 0, 9, 0, 13, 0, 23, 0, 34, 0, 54, 0, 78, 0, 120, 0, 170, 0, 252, 0, 358, 0, 517, 0, 725, 0, 1030, 0, 1427, 0, 1992, 0, 2733, 0, 3759, 0, 5106, 0, 6946, 0, 9345, 0, 12577, 0, 16788, 0, 22384, 0, 29641, 0

Offset: 0

Author

Gus Wiseman, Oct 01 2022

Keywords

• nonn

Comments

A partition of n is a weakly decreasing sequence of positive integers summing to n.

The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^i y_i.

Examples

			The a(1) = 1 through a(13) = 9 partitions:
  1   .  .  .  311   .  322   .  333     .  443     .  553
                        421      432        542        652
                                 531        641        751
                                 51111      52211      52222
                                            62111      53311
                                                       62221
                                                       63211
                                                       73111
                                                       7111111
		

Crossrefs

For product equal to sum we have A001055, compositions A335405.

The version for compositions is A357182, reverse ranked by A357184.

The reverse version is A357189, ranked by A357486.

These partitions are ranked by A357485.

Removing zeros gives A357488.

A000041 counts partitions, strict A000009.

A025047 counts alternating compositions.

A103919 counts partitions by alternating sum, full triangle A344651.

A357136 counts compositions by alternating sum, full triangle A097805.

Cf. A004526, A051159, A114220, A131044, A262046, A262977, A301987, A357183.

Programs

Mathematica

ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Table[Length[Select[IntegerPartitions[n],Length[#]==ats[Reverse[#]]&]],{n,0,30}]