OFFSET
0,4
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(7) = 12 partitions:
(2) (3) (4) (5) (6) (7)
(21) (31) (32) (42) (43)
(211) (41) (51) (52)
(311) (222) (61)
(2111) (321) (322)
(411) (421)
(3111) (511)
(21111) (2221)
(3211)
(4111)
(31111)
(211111)
MAPLE
a:= n-> (p-> p(n)-p(iquo(n, 2)))(combinat[numbpart]):
seq(a(n), n=0..63); # Alois P. Heinz, Oct 04 2021
MATHEMATICA
altprod[q_]:=Product[q[[i]]^(-1)^(i-1), {i, Length[q]}];
Table[Length[Select[IntegerPartitions[n], altprod[#]>1&]], {n, 0, 30}]
CROSSREFS
The strict case is A000009, except that a(0) = a(1) = 0.
The even-length case is A344608.
Allowing any integer reverse-alternating product gives A347445.
Allowing any integer alternating product gives A347446.
The reverse version is A347449, also the odd-length case.
A027187 counts partitions of even length.
A027193 counts partitions of odd length.
A347461 counts possible alternating products of partitions.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 16 2021
STATUS
approved