OFFSET
0,9
COMMENTS
The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2).
The zeroth differences of a sequence are the sequence itself, while the k-th differences for k > 0 are the differences of the (k-1)-th differences. The distinct differences of any degree are the union of the k-th differences for all k >= 0. For example, the k-th differences of (1,1,2,4) for k = 0...3 are:
(1,1,2,4)
(0,1,2)
(1,1)
(0)
so there are a total of 4 distinct differences of any degree, namely {0,1,2,4}.
LINKS
EXAMPLE
Triangle begins:
1
0 1
0 1 1
0 1 2 0
0 1 2 2 0
0 1 1 3 2 0
0 1 4 2 3 1 0
0 1 1 5 5 2 1 0
0 1 3 5 6 3 3 1 0
0 1 3 4 8 7 1 4 2 0
0 1 3 6 11 7 5 2 4 2 1
0 1 1 6 13 8 9 9 0 4 3 1
0 1 6 7 11 12 9 10 8 4 3 2 2
0 1 1 7 18 9 14 19 5 10 3 5 4 1
0 1 3 9 17 9 22 20 15 9 7 6 5 4 1
0 1 4 8 22 11 16 24 22 19 10 11 2 8 7 2
0 1 4 10 23 15 24 23 27 27 12 14 11 8 8 5 5
Row n = 8 counts the following partitions:
(8) (44) (17) (116) (134) (1133) (111122)
(2222) (26) (125) (233) (11123)
(11111111) (35) (1115) (1223) (11222)
(224) (1124)
(1111112) (11114)
(111113)
MATHEMATICA
Table[Length[Select[Reverse/@IntegerPartitions[n], Length[Union@@Table[Differences[#, i], {i, 0, Length[#]}]]==k&]], {n, 0, 16}, {k, 0, n}]
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, May 03 2019
STATUS
approved