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A349061
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Number of integer partitions of n with at least one part of odd multiplicity that is not the first or last.
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23
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0, 0, 0, 0, 0, 0, 1, 2, 4, 8, 13, 21, 32, 48, 67, 99, 133, 185, 245, 333, 432, 574, 732, 957, 1208, 1554, 1941, 2468, 3060, 3844, 4731, 5893, 7204, 8898, 10816, 13268, 16043, 19546, 23523, 28497, 34150, 41147, 49106, 58892, 70020, 83597, 99047, 117778, 139087
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OFFSET
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0,8
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COMMENTS
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Also the number of non-weakly alternating integer partitions of n, where we define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. This sequence looks at the somewhat degenerate case where no strict increases are allowed.
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LINKS
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EXAMPLE
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The a(6) = 1 through a(10) = 13 partitions:
(321) (421) (431) (432) (532)
(3211) (521) (531) (541)
(4211) (621) (631)
(32111) (3321) (721)
(4311) (4321)
(5211) (5311)
(42111) (6211)
(321111) (32221)
(33211)
(43111)
(52111)
(421111)
(3211111)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], !SameQ@@#&&!And@@EvenQ/@Take[Length/@Split[#], {2, -2}]&]], {n, 0, 30}]
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CROSSREFS
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The complement is counted by A349060.
These partitions are ranked by A349794.
A003242 counts Carlitz (anti-run) compositions.
A096441 counts weakly alternating 0-appended partitions.
A345170 counts partitions w/ an alternating permutation, ranked by A345172.
A349052 counts weakly alternating compositions.
A349056 counts weakly alternating permutations of prime indices.
A349798 counts weakly but not strongly alternating perms of prime indices.
Cf. A002865, A027383, A117298, A117989, A129852, A129853, A274230, A333213, A344615, A345165, A349059.
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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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