A308230 Irregular triangle: row n shows the alternating sums of partitions of n when the parts are arranged in nonincreasing order and the partitions are arranged lexicographically from [n] to [1,1,1,...,1].

1, 2, 0, 3, 1, 1, 4, 2, 0, 2, 0, 5, 3, 1, 3, 1, 1, 1, 6, 4, 2, 4, 0, 2, 2, 2, 0, 2, 0, 7, 5, 3, 5, 1, 3, 3, 1, 3, 1, 3, 1, 1, 1, 1, 8, 6, 4, 6, 2, 4, 4, 0, 2, 4, 2, 4, 2, 0, 2, 2, 2, 0, 2, 0, 2, 0, 9, 7, 5, 7, 3, 5, 5, 1, 3, 5, 3, 5, 1, 3, 1, 3, 3, 3, 3, 1

Offset: 1

Author

Clark Kimberling, May 17 2019

Keywords

• nonn
• tabf

Comments

Row n consists of A000041(n) numbers, for n >= 1. The numbers in row n have the parity of n. Regarding row sums, see Comments at A066897.

Examples

			First 8 rows:
  1
  2  0
  3  1  1
  4  2  0  2  0
  5  3  1  3  1  1  1
  6  4  2  4  0  2  2  2  0  2  0
  7  5  3  5  1  3  3  1  3  1  3  1  1  1  1
  8  6  4  6  2  4  4  0  2  4  2  4  2  0  2  2  2  0  2  0  2  0
Row 5 comes from arranging the 7 partitions of 5 is this order:
[5], [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1], [1,1,1,1,1] and taking alternating sums: 5, 4-1, 3-1, 3-1+1, 2-2+1, 2-1+1-1, 1-1+1-1.
		

Crossrefs

Cf. A000041, A066897.

Programs

Mathematica

r[n_] := Map[Total[Map[Total, {Take[##], Drop[##]} &[#, {1, -1, 2}] {1, -1}]] &, IntegerPartitions[n]]; Column[Table[r[n], {n, 10}]] (* Peter J. C. Moses, May 15 2019 *)