OFFSET
0,8
COMMENTS
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 + ...
EXAMPLE
Triangle begins:
1
0 1
0 1 1
0 2 1 1
0 3 3 1 1
0 4 5 5 1 1
0 5 7 10 8 1 1
0 6 9 17 18 12 1 1
0 7 11 27 35 29 17 1 1
0 8 13 41 63 63 43 23 1 1
0 9 15 60 106 126 104 60 30 1 1
Row n = 6 counts the following compositions:
(15) (24) (33) (42) (51) (6)
(114) (213) (312) (411)
(123) (222) (321) (1113)
(132) (231) (1122) (2112)
(141) (1131) (1212) (3111)
(1221) (2121) (11112)
(1311) (2211) (11121)
(11211) (21111)
(12111)
(111111)
MATHEMATICA
skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]), {i, Length[f]}];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], skats[#]==k&]], {n, 0, 10}, {k, -n, n, 2}]
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Oct 12 2022
STATUS
approved