OFFSET
0,4
COMMENTS
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.
FORMULA
EXAMPLE
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)
MATHEMATICA
altprod[q_]:=Product[q[[i]]^(-1)^(i-1), {i, Length[q]}];
Table[Length[Select[IntegerPartitions[n], altprod[Reverse[#]]<=1&]], {n, 0, 30}]
CROSSREFS
The odd-length case is A035363 (shifted).
The strict case is A067661.
The even bisection is A236913.
The opposite version (>= instead of <=) is A344607.
The case of < 1 instead of <= 1 is A344608.
Allowing any integer reverse-alternating product gives A347445.
The complement (> 1 instead of <= 1) is counted by A347449.
A000041 counts partitions.
A027187 counts partitions of even length.
A027193 counts partitions of odd length.
A100824 counts partitions with alternating sum <= 1.
A347461 counts possible alternating products of partitions.
A347462 counts possible reverse-alternating products of partitions.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 14 2021
STATUS
approved