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A347704
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Number of even-length integer partitions of n with integer alternating product.
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6
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1, 0, 1, 1, 3, 2, 6, 4, 11, 8, 18, 13, 33, 22, 49, 38, 79, 58, 122, 90, 186, 139, 268, 206, 402, 304, 569, 448, 817, 636, 1152, 907, 1612, 1283, 2220, 1791, 3071, 2468, 4162, 3409, 5655, 4634, 7597, 6283, 10171, 8478, 13491, 11336, 17906, 15088, 23513, 20012
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OFFSET
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0,5
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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)).
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LINKS
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EXAMPLE
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The a(2) = 1 through a(9) = 8 partitions:
(11) (21) (22) (41) (33) (61) (44) (63)
(31) (2111) (42) (2221) (62) (81)
(1111) (51) (4111) (71) (3321)
(2211) (211111) (2222) (4221)
(3111) (3221) (6111)
(111111) (3311) (222111)
(4211) (411111)
(5111) (21111111)
(221111)
(311111)
(11111111)
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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], EvenQ[Length[#]]&&IntegerQ[altprod[#]]&]], {n, 0, 30}]
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CROSSREFS
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Allowing any alternating product gives A027187, odd bisection A236914.
The Heinz numbers of these partitions are given by A028260 /\ A347457.
The reverse and reciprocal versions are both A035363.
The multiplicative version (factorizations) is A347438, reverse A347439.
The odd-length instead of even-length version is A347444.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A119620 counts partitions with alternating product 1.
Cf. A000070, A067661, A236913, A304620, A339846, A347437, A347441, A347442, A347445, A347448, A347449, A347454, A347462.
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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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