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A325466
Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with k distinct differences of any degree > 0.
5
1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 1, 0, 0, 1, 3, 2, 1, 0, 0, 1, 5, 4, 0, 1, 0, 0, 1, 4, 6, 3, 0, 1, 0, 0, 1, 6, 6, 4, 3, 1, 1, 0, 0, 1, 6, 10, 4, 2, 4, 1, 2, 0, 0, 1, 7, 12, 8, 3, 3, 4, 1, 2, 1, 0, 1, 6, 13, 11, 2, 11, 3, 4, 0, 3, 1, 1, 1, 10, 16, 7, 10, 10
OFFSET
0,8
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.
EXAMPLE
Triangle begins:
1
1 0
1 1 0
1 2 0 0
1 3 1 0 0
1 3 2 1 0 0
1 5 4 0 1 0 0
1 4 6 3 0 1 0 0
1 6 6 4 3 1 1 0 0
1 6 10 4 2 4 1 2 0 0
1 7 12 8 3 3 4 1 2 1 0
1 6 13 11 2 11 3 4 0 3 1 1
1 10 16 7 10 10 6 6 5 1 1 2 1
1 7 18 14 7 16 11 6 4 8 0 5 0 1
1 9 20 18 10 20 13 10 10 4 5 5 2 2 2
1 10 26 18 10 24 13 19 13 10 6 6 2 8 1 2
1 11 25 24 16 28 19 24 14 15 9 10 9 5 2 7 1
Row 7 counts the following reversed partitions (empty columns not shown):
(7) (16) (115) (133) (11122)
(25) (124) (1123)
(34) (223) (1222)
(1111111) (1114)
(11113)
(111112)
Row 9 counts the following reversed partitions (empty columns not shown):
(9) (18) (117) (126) (1125) (1134) (11223) (111222)
(27) (135) (144) (11124) (1224) (1111122)
(36) (225) (1233) (11133)
(45) (234) (12222) (111123)
(333) (1116)
(111111111) (2223)
(11115)
(111114)
(1111113)
(11111112)
MATHEMATICA
Table[Length[Select[Reverse/@IntegerPartitions[n], Length[Union@@Table[Differences[#, i], {i, 1, Length[#]}]]==k&]], {n, 0, 16}, {k, 0, n}]
CROSSREFS
Row sums are A000041. Column k = 1 is A088922.
Sequence in context: A029362 A216599 A114510 * A077029 A052553 A290054
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, May 04 2019
STATUS
approved