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 0 1
0 2 2 0 1
0 3 1 2 0 1
0 3 2 3 2 0 1
0 4 2 4 1 3 0 1
0 4 3 3 6 2 3 0 1
0 5 3 5 3 7 2 4 0 1
0 5 4 5 4 9 7 3 4 0 1
0 6 4 7 3 12 5 10 3 5 0 1
0 6 5 7 5 10 16 7 11 4 5 0 1
0 7 5 9 5 14 11 18 7 14 4 6 0 1
Row n = 7 counts the following reversed partitions:
. (16) (25) (34) (1123) (1114) . (7)
(115) (223) (1222) (11113)
(124) (111112) (11122)
(133) (1111111)
MATHEMATICA
skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]), {i, Length[f]}];
Table[Length[Select[Reverse/@IntegerPartitions[n], skats[#]==k&]], {n, 0, 11}, {k, -n, n, 2}]
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Oct 10 2022
STATUS
approved