OFFSET
0,13
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 + ....
FORMULA
Conjecture: The columns are palindromes with sums A298311.
EXAMPLE
Triangle begins:
1
0 1
0 1 1
0 1 1 1
0 0 3 1 1
0 0 1 4 1 1
0 0 1 4 4 1 1
0 0 0 4 5 4 1 1
0 0 0 1 10 5 4 1 1
0 0 0 1 5 13 5 4 1 1
0 0 0 0 4 13 14 5 4 1 1
0 0 0 0 1 13 17 14 5 4 1 1
0 0 0 0 1 5 28 18 14 5 4 1 1
Row n = 7 counts the following partitions:
. . . (322) (43) (52) (61) (7)
(331) (421) (511)
(2221) (3211) (4111)
(1111111) (22111) (31111)
(211111)
MATHEMATICA
skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]), {i, Length[f]}];
Table[Length[Select[IntegerPartitions[n], skats[#]==k&]], {n, 0, 12}, {k, -n, n, 2}]
CROSSREFS
Row sums are A000041.
Number of nonzero entries in row n appears to be A004396(n+1).
First nonzero entry of each row appears to converge to A146325.
Column sums appear to be A298311.
The half-alternating version is A357637.
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Oct 10 2022
STATUS
approved