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A325874
Number of integer partitions of n whose differences of all degrees > 1 are nonzero.
10
1, 1, 2, 2, 4, 5, 6, 8, 12, 13, 19, 24, 26, 33, 45, 52, 66, 78, 92, 113, 129, 160, 192, 231, 268, 305, 361, 436, 501, 591, 665, 783, 897, 1071, 1228, 1361, 1593, 1834, 2101, 2452, 2685, 3129, 3526, 4067, 4568, 5189, 5868, 6655, 7565, 8468, 9400
OFFSET
0,3
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 are the sequence itself, while k-th differences for k > 0 are the differences of the (k-1)-th differences. If m is the length of the sequence, its differences of all degrees are the union of the zeroth through m-th differences.
The case for all degrees including 1 is A325852.
EXAMPLE
The a(1) = 1 through a(9) = 13 partitions:
(1) (2) (3) (4) (5) (6) (7) (8) (9)
(11) (21) (22) (32) (33) (43) (44) (54)
(31) (41) (42) (52) (53) (63)
(211) (221) (51) (61) (62) (72)
(311) (411) (322) (71) (81)
(2211) (331) (332) (441)
(421) (422) (522)
(511) (431) (621)
(521) (711)
(611) (4221)
(3221) (4311)
(3311) (5211)
(32211)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], !MemberQ[Union@@Table[Differences[#, i], {i, 2, Length[#]}], 0]&]], {n, 0, 30}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 02 2019
STATUS
approved