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A347704
Number of even-length integer partitions of n with integer alternating product.
6
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
OFFSET
0,5
COMMENTS
We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)).
EXAMPLE
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)
MATHEMATICA
altprod[q_]:=Product[q[[i]]^(-1)^(i-1), {i, Length[q]}];
Table[Length[Select[IntegerPartitions[n], EvenQ[Length[#]]&&IntegerQ[altprod[#]]&]], {n, 0, 30}]
CROSSREFS
Allowing any alternating product >= 1 gives A000041, reverse A344607.
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.
Allowing any length gives A347446.
A034008 counts even-length compositions, ranked by A053754.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A119620 counts partitions with alternating product 1.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
Sequence in context: A329435 A160795 A258212 * A355019 A092401 A222208
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 17 2021
STATUS
approved