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.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 09 2020
STATUS
approved