OFFSET
0,4
COMMENTS
Conjecture: a(n) >= A236914.
The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i. This is equal to (-1)^(m-1) times the number of odd parts in the conjugate partition, where m is the number of parts. So a(n) is the number of even-length partitions of 2n with at least one odd conjugate part. By conjugation, this is also the number of partitions of 2n with greatest part even and at least one odd part.
The alternating sum of a partition is never < 0, so the non-reverse version is A000004.
EXAMPLE
The a(2) = 1 through a(5) = 15 partitions:
(31) (42) (53) (64)
(51) (62) (73)
(3111) (71) (82)
(3221) (91)
(4211) (3331)
(5111) (4222)
(311111) (4321)
(5221)
(5311)
(6211)
(7111)
(322111)
(421111)
(511111)
(31111111)
MATHEMATICA
sats[y_] := Sum[(-1)^(i - Length[y])*y[[i]], {i, Length[y]}];
Table[Length[Select[IntegerPartitions[n], sats[#]<0&]], {n, 0, 30, 2}]
CROSSREFS
The ordered version (compositions not partitions) appears to be A008549.
Even bisection of A344608.
The complementary partitions of 2n are counted by A344611.
A344610 counts partitions by sum and positive reverse-alternating sum.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 09 2021
EXTENSIONS
More terms from Bert Dobbelaere, Jun 12 2021
STATUS
approved