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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
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
1,2
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.
EXAMPLE
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.
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 *)
CROSSREFS
Sequence in context: A362557 A047983 A070812 * A061865 A135818 A078804
KEYWORD
nonn,tabf
AUTHOR
Clark Kimberling, May 17 2019
STATUS
approved