OFFSET
0,6
COMMENTS
We define the half-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 0 2
0 0 1 3
0 0 2 2 4
0 0 3 5 3 5
0 0 4 8 10 4 6
0 0 5 11 18 18 5 7
0 0 6 14 28 36 30 6 8
0 0 7 17 41 63 65 47 7 9
0 0 8 20 58 104 126 108 70 8 10
Row n = 6 counts the following compositions:
(114) (123) (132) (141) (6)
(1113) (213) (222) (231) (15)
(1122) (1212) (312) (321) (24)
(1131) (1221) (1311) (411) (33)
(2112) (2211) (42)
(2121) (3111) (51)
(11121) (11112)
(11211) (12111)
(21111)
(111111)
MATHEMATICA
halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]), {i, Length[f]}];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], halfats[#]==k&]], {n, 0, 10}, {k, -n, n, 2}]
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Oct 12 2022
STATUS
approved