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A336106
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Number of integer partitions of n whose greatest part is at most one more than the sum of the other parts.
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2
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1, 1, 1, 2, 3, 5, 7, 11, 15, 23, 30, 44, 58, 82, 105, 146, 186, 252, 318, 423, 530, 695, 863, 1116, 1380, 1763, 2164, 2738, 3345, 4192, 5096, 6334, 7665, 9459, 11395, 13968, 16765, 20425, 24418, 29588, 35251, 42496, 50460, 60547, 71669, 85628
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OFFSET
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0,4
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COMMENTS
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Also the number of separable strong multisets of length n covering an initial interval of positive integers. A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts.
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LINKS
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EXAMPLE
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The a(1) = 1 through a(8) = 15 partitions:
(1) (11) (21) (22) (32) (33) (43) (44)
(111) (211) (221) (222) (322) (332)
(1111) (311) (321) (331) (422)
(2111) (2211) (421) (431)
(11111) (3111) (2221) (2222)
(21111) (3211) (3221)
(111111) (4111) (3311)
(22111) (4211)
(31111) (22211)
(211111) (32111)
(1111111) (41111)
(221111)
(311111)
(2111111)
(11111111)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], 2*Max@@#<=1+n&]], {n, 0, 15}]
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CROSSREFS
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The inseparable version is A025065.
The Heinz numbers of these partitions are A335127.
Sequences covering an initial interval are A000670.
Inseparable partitions are A325535.
Separable factorizations are A335434.
Heinz numbers of separable partitions are A335433.
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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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