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 2
0 0 2 0 3
0 0 2 2 0 3
0 0 3 1 3 0 4
0 0 3 2 4 2 0 4
0 0 4 2 6 2 3 0 5
0 0 4 3 5 7 3 3 0 5
0 0 5 3 8 4 10 2 4 0 6
0 0 5 4 8 6 11 9 3 4 0 6
0 0 6 4 11 5 15 8 13 3 5 0 7
0 0 6 5 11 8 13 19 10 13 4 5 0 7
0 0 7 5 14 8 19 13 25 9 17 4 6 0 8
0 0 7 6 14 11 19 17 29 23 13 18 5 6 0 8
Row n = 7 counts the following reversed partitions:
. . (115) (124) (133) (11113) . (7)
(1114) (1222) (223) (111112) (16)
(1123) (11122) (25)
(1111111) (34)
MATHEMATICA
halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]), {i, Length[f]}];
Table[Length[Select[Reverse/@IntegerPartitions[n], halfats[#]==k&]], {n, 0, 15}, {k, -n, n, 2}]
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Oct 10 2022
STATUS
approved