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A325874
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Number of integer partitions of n whose differences of all degrees > 1 are nonzero.
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10
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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
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], !MemberQ[Union@@Table[Differences[#, i], {i, 2, Length[#]}], 0]&]], {n, 0, 30}]
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CROSSREFS
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Cf. A049988, A238423, A325325, A325468, A325545, A325849, A325850, A325851, A325852, A325875, A325876.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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