OFFSET
0,4
COMMENTS
The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i. This is equal to (-1)^(r-1) times the number of odd parts, where r is the greatest part, so a(n) is the number of integer partitions of 2n with exactly two odd parts, neither of which is the greatest.
Also the number of reversed integer partitions of 2n with alternating sum -2.
EXAMPLE
The a(2) = 1 through a(6) = 14 partitions:
(31) (42) (53) (64) (75)
(3111) (3221) (3331) (4332)
(4211) (4222) (4431)
(311111) (4321) (5322)
(5311) (5421)
(322111) (6411)
(421111) (322221)
(31111111) (333111)
(422211)
(432111)
(531111)
(32211111)
(42111111)
(3111111111)
MATHEMATICA
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]], {i, Length[y]}];
Table[Length[Select[IntegerPartitions[n], sats[#]==-2&]], {n, 0, 30, 2}]
- or -
Table[Length[Select[IntegerPartitions[n], EvenQ[Max[#]]&&Count[#, _?OddQ]==2&]], {n, 0, 30, 2}]
CROSSREFS
The version for -1 instead of -2 is A000070.
The non-reversed negative version is A000097.
The ordered version appears to be A001700.
The version for 1 instead of -2 is A035363.
The whole set of partitions of 2n is counted by A058696.
The strict case appears to be A065033.
The version for -1 instead of -2 is A306145.
The version for 2 instead of -2 is A344613.
A344610 counts partitions by sum and positive reverse-alternating sum.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 08 2021
EXTENSIONS
More terms from Bert Dobbelaere, Jun 12 2021
STATUS
approved