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A336106
Number of integer partitions of n whose greatest part is at most one more than the sum of the other parts.
2
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
OFFSET
0,4
COMMENTS
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.
EXAMPLE
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)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], 2*Max@@#<=1+n&]], {n, 0, 15}]
CROSSREFS
The inseparable version is A025065.
The Heinz numbers of these partitions are A335127.
The non-strong version is A336103.
Sequences covering an initial interval are A000670.
Anti-run compositions are A003242.
Anti-run patterns are A005649.
Separable partitions are A325534.
Inseparable partitions are A325535.
Separable factorizations are A335434.
Heinz numbers of separable partitions are A335433.
Sequence in context: A353723 A308990 A321142 * A366845 A024792 A280661
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 09 2020
STATUS
approved