OFFSET
0,6
COMMENTS
The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.
Also the number of reversed of integer partitions of n with alternating sum < 0.
No integer partitions have alternating sum < 0, so the non-reversed version is all zeros.
A formula for the reverse-alternating sum of a partition is: (-1)^(k-1) times the number of odd parts in the conjugate partition, where k is the number of parts. So a(n) is the number of integer partitions of n of even length whose conjugate parts are not all odd. Partitions of the latter type are counted by A086543. By conjugation, a(n) is also the number of integer partitions of n of even maximum whose parts are not all odd.
EXAMPLE
The a(3) = 1 through a(9) = 14 partitions:
(21) (31) (32) (42) (43) (53) (54)
(41) (51) (52) (62) (63)
(2111) (3111) (61) (71) (72)
(2221) (3221) (81)
(3211) (4211) (3222)
(4111) (5111) (3321)
(211111) (311111) (4221)
(4311)
(5211)
(6111)
(222111)
(321111)
(411111)
(21111111)
MATHEMATICA
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]], {i, Length[y]}];
Table[Length[Select[IntegerPartitions[n], sats[#]<0&]], {n, 0, 30}]
CROSSREFS
A344604 counts wiggly compositions with twins.
A344610 counts partitions by sum and positive reverse-alternating sum.
A344618 gives reverse-alternating sums of standard compositions.
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 30 2021
STATUS
approved