OFFSET
0,5
COMMENTS
From Gus Wiseman, Oct 12 2022: (Start)
Also the number of integer partitions of n whose skew-alternating sum is 0, where we define the skew-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A - B - C + D + E - F - G + ... These are the conjugates of the partitions described in the name. For example, the a(0) = 1 through a(8) = 10 partitions are:
() . (11) . (22) . (33) . (44)
(211) (321) (422)
(1111) (2211) (431)
(111111) (2222)
(3221)
(3311)
(22211)
(221111)
(2111111)
(11111111)
The ordered version (compositions) is A138364
The reverse version is A357640 (aerated).
(End)
EXAMPLE
From Gus Wiseman, Oct 12 2022: (Start)
The a(0) = 1 through a(8) = 10 partitions:
() . (2) . (4) . (6) . (8)
(22) (42) (44)
(31) (222) (53)
(321) (62)
(71)
(422)
(431)
(2222)
(3221)
(3311)
(End)
MATHEMATICA
skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]), {i, Length[f]}];
Table[Length[Select[IntegerPartitions[n], skats[#]==0&]], {n, 0, 30}] (* Gus Wiseman, Oct 12 2022 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from David W. Wilson
STATUS
approved