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A389920
Number of integer partitions of n not forming an arithmetic progression with offset 0.
2
0, 0, 1, 1, 4, 6, 8, 14, 21, 28, 40, 55, 74, 100, 134, 173, 230, 296, 382, 489, 625, 789, 1001, 1254, 1572, 1957, 2435, 3008, 3716, 4564, 5599, 6841, 8348, 10141, 12309, 14882, 17973, 21636, 26014, 31183, 37336, 44582, 53170, 63260, 75174, 89130, 105557
OFFSET
0,5
COMMENTS
These are partitions whose 0-appended first differences are not all equal.
FORMULA
a(n) = A000041(n) - A007862(n).
EXAMPLE
The partition y = (3,2,2,1) has 0-appended differences (-1,0,-1,-1), which are not all equal, so y is counted under a(8).
The a(2) = 1 through a(8) = 21 partitions:
(11) (111) (22) (32) (33) (43) (44)
(31) (41) (51) (52) (53)
(211) (221) (222) (61) (62)
(1111) (311) (411) (322) (71)
(2111) (2211) (331) (332)
(11111) (3111) (421) (422)
(21111) (511) (431)
(111111) (2221) (521)
(3211) (611)
(4111) (2222)
(22111) (3221)
(31111) (3311)
(211111) (4211)
(1111111) (5111)
(22211)
(32111)
(41111)
(221111)
(311111)
(2111111)
(11111111)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], !SameQ@@Differences[Append[#, 0]]&]], {n, 0, 15}]
CROSSREFS
The complement is counted by A007862, ranks A325327.
The non 0-appended complement is A049988, ranks A325328.
The complement for distinct instead of equal differences A325324.
For compositions instead of partitions we have A389742, ranks A389736.
The non 0-appended version is A389811, ranks A389812.
A000041 counts integer partitions, strict A000009.
A000837 counts aperiodic partitions.
A175342 counts compositions with equal differences, ranks A389731, subsets A051336.
A389741 counts compositions without equal differences, ranks A389735.
Sequence in context: A116897 A384662 A293763 * A246324 A280227 A181978
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 21 2025
STATUS
approved