OFFSET
0,4
COMMENTS
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.
EXAMPLE
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)
MATHEMATICA
altprod[q_]:=Product[q[[i]]^(-1)^(i-1), {i, Length[q]}];
Table[Length[Select[IntegerPartitions[n], OddQ[Length[#]]&&IntegerQ[altprod[#]]&]], {n, 0, 30}]
CROSSREFS
The reciprocal version is A035363.
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.
Ranked by A347453.
The even-length instead of odd-length version is A347704.
A000041 counts partitions.
A025047 counts wiggly compositions.
A026424 lists numbers with odd bigomega.
A339890 counts odd-length factorizations.
A347437 counts factorizations with integer alternating product.
A347461 counts possible alternating products of partitions.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 14 2021
STATUS
approved