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A347445
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Number of integer partitions of n with integer reverse-alternating product.
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20
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1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 20, 24, 32, 40, 50, 62, 77, 99, 115, 151, 170, 224, 251, 331, 360, 481, 517, 690, 728, 980, 1020, 1379, 1420, 1918, 1962, 2643, 2677, 3630, 3651, 4920, 4926, 6659, 6625, 8931, 8853, 11905, 11781, 15805, 15562, 20872, 20518
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OFFSET
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0,3
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COMMENTS
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We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)). The reverse-alternating product is the alternating product of the reversed sequence.
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LINKS
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EXAMPLE
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The a(1) = 1 through a(8) = 12 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (111) (22) (221) (33) (322) (44)
(211) (311) (222) (331) (332)
(1111) (11111) (411) (421) (422)
(2211) (511) (611)
(21111) (22111) (2222)
(111111) (31111) (3311)
(1111111) (22211)
(41111)
(221111)
(2111111)
(11111111)
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MATHEMATICA
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revaltprod[q_]:=Product[Reverse[q][[i]]^(-1)^(i-1), {i, Length[q]}];
Table[Length[Select[IntegerPartitions[n], IntegerQ[revaltprod[#]]&]], {n, 0, 30}]
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CROSSREFS
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Allowing any reverse-alternating product >= 1 gives A344607.
Allowing any reverse-alternating product < 1 gives A344608.
Allowing any reverse-alternating product <= 1 gives A347443.
Allowing any reverse-alternating product > 1 gives A347449.
A027187 counts partitions of even length.
A027193 counts partitions of odd length.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A339890 counts factorizations with alternating product > 1, reverse A347705.
A347462 counts possible reverse-alternating products of partitions.
Cf. A025047, A067661, A119620, A344654, A344740, A347439, A347440, A347448, A347450, A347451, A347461, A347463, A347704.
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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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