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 - ...
LINKS
Alois P. Heinz, Rows n = 0..200, flattened
FORMULA
Conjecture: The column sums are A029862.
EXAMPLE
Triangle begins:
1
0 1
0 0 2
0 0 1 2
0 0 1 1 3
0 0 0 2 2 3
0 0 0 0 5 2 4
0 0 0 0 2 6 3 4
0 0 0 0 2 3 9 3 5
0 0 0 0 0 4 7 10 4 5
0 0 0 0 0 0 11 8 13 4 6
0 0 0 0 0 0 4 15 12 14 5 6
0 0 0 0 0 0 3 7 25 13 17 5 7
Row n = 9 counts the following partitions:
(3222) (333) (432) (441) (9)
(22221) (3321) (522) (531) (54)
(21111111) (4221) (4311) (621) (63)
(111111111) (32211) (5211) (711) (72)
(222111) (6111) (81)
(2211111) (33111)
(3111111) (42111)
(51111)
(321111)
(411111)
MAPLE
b:= proc(n, i, s, t) option remember; `if`(n=0, x^s, `if`(i<1, 0,
b(n, i-1, s, t)+b(n-i, min(n-i, i), s+`if`(t<2, i, -i), irem(t+1, 4))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=-n..n, 2))(b(n$2, 0$2)):
seq(T(n), n=0..15); # Alois P. Heinz, Oct 12 2022
MATHEMATICA
halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]), {i, Length[f]}];
Table[Length[Select[IntegerPartitions[n], halfats[#]==k&]], {n, 0, 12}, {k, -n, n, 2}]
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Oct 10 2022
STATUS
approved