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A347443
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Number of integer partitions of n with reverse-alternating product <= 1.
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17
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1, 1, 1, 2, 3, 5, 6, 10, 12, 19, 22, 34, 40, 60, 69, 101, 118, 168, 195, 272, 317, 434, 505, 679, 793, 1050, 1224, 1599, 1867, 2409, 2811, 3587, 4186, 5290, 6168, 7724, 9005, 11186, 13026, 16062, 18692, 22894, 26613, 32394, 37619, 45535, 52815, 63593, 73680
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OFFSET
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0,4
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COMMENTS
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Includes all partitions of even length (A027187).
Also the number of integer partitions of n with reverse-alternating sum <= 1.
Also the number of integer partitions of n having either even length (A027187) or having exactly one odd part in the conjugate partition (A100824).
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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FORMULA
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EXAMPLE
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The a(1) = 1 through a(8) = 12 partitions:
(1) (11) (21) (22) (32) (33) (43) (44)
(111) (31) (41) (42) (52) (53)
(1111) (221) (51) (61) (62)
(2111) (2211) (331) (71)
(11111) (3111) (2221) (2222)
(111111) (3211) (3221)
(4111) (3311)
(22111) (4211)
(211111) (5111)
(1111111) (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], altprod[Reverse[#]]<=1&]], {n, 0, 30}]
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CROSSREFS
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The odd-length case is A035363 (shifted).
The opposite version (>= instead of <=) is A344607.
The case of < 1 instead of <= 1 is A344608.
The multiplicative version (factorizations) is A347438, non-reverse A339846.
Allowing any integer reverse-alternating product gives A347445.
The complement (> 1 instead of <= 1) is counted by A347449.
A027187 counts partitions of even length.
A027193 counts partitions of odd length.
A058622 counts compositions with alternating sum <= 0 (A294175 for < 0).
A100824 counts partitions with alternating sum <= 1.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A347461 counts possible alternating products of partitions.
A347462 counts possible reverse-alternating products of partitions.
Cf. A000070, A038548, A086543, A116406, A325534, A325535, A344611, A344654, A344740, A347440, A347442, A347446, A347448.
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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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