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A347444
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Number of odd-length integer partitions of n with integer alternating product.
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13
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0, 1, 1, 2, 2, 4, 4, 8, 7, 14, 13, 24, 21, 40, 35, 62, 55, 99, 85, 151, 128, 224, 195, 331, 283, 481, 416, 690, 593, 980, 844, 1379, 1189, 1918, 1665, 2643, 2292, 3630, 3161, 4920, 4299, 6659, 5833, 8931, 7851, 11905, 10526, 15805, 13987, 20872, 18560, 27398
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OFFSET
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0,4
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COMMENTS
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We define the alternating product of a sequence (y_1, ... ,y_k) to be the Product_i y_i^((-1)^(i-1)).
The reverse version (integer reverse-alternating product) is the same.
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LINKS
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EXAMPLE
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The a(1) = 1 through a(9) = 14 partitions:
(1) (2) (3) (4) (5) (6) (7) (8) (9)
(111) (211) (221) (222) (322) (332) (333)
(311) (411) (331) (422) (441)
(11111) (21111) (421) (611) (522)
(511) (22211) (621)
(22111) (41111) (711)
(31111) (2111111) (22221)
(1111111) (32211)
(33111)
(42111)
(51111)
(2211111)
(3111111)
(111111111)
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MATHEMATICA
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altprod[q_]:=Product[q[[i]]^(-1)^(i-1), {i, Length[q]}];
Table[Length[Select[IntegerPartitions[n], OddQ[Length[#]]&&IntegerQ[altprod[#]]&]], {n, 0, 30}]
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CROSSREFS
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Allowing any alternating product gives A027193.
The multiplicative version (factorizations) is A347441.
Allowing any length and alternating product > 1 gives A347448.
Allowing any reverse-alternating product > 1 gives A347449.
The even-length instead of odd-length version is A347704.
A025047 counts wiggly compositions.
A026424 lists numbers with odd bigomega.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A119620 counts partitions with alternating product 1, ranked by A028982.
A339890 counts odd-length factorizations.
A347437 counts factorizations with integer alternating product.
A347461 counts possible alternating products of partitions.
Cf. A000070, A236559, A236913, A236914, A304620, A344654, A347439, A347442, A347456, A347457, A347460, A347462, A347463.
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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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