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A357646
Triangle read by rows where T(n,k) is the number of integer compositions of n with skew-alternating sum k, where k ranges from -n to n in steps of 2.
8
1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 3, 3, 1, 1, 0, 4, 5, 5, 1, 1, 0, 5, 7, 10, 8, 1, 1, 0, 6, 9, 17, 18, 12, 1, 1, 0, 7, 11, 27, 35, 29, 17, 1, 1, 0, 8, 13, 41, 63, 63, 43, 23, 1, 1, 0, 9, 15, 60, 106, 126, 104, 60, 30, 1, 1, 0, 10, 17, 85, 168, 232, 230, 162, 80, 38, 1, 1
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 1 1
0 3 3 1 1
0 4 5 5 1 1
0 5 7 10 8 1 1
0 6 9 17 18 12 1 1
0 7 11 27 35 29 17 1 1
0 8 13 41 63 63 43 23 1 1
0 9 15 60 106 126 104 60 30 1 1
Row n = 6 counts the following compositions:
(15) (24) (33) (42) (51) (6)
(114) (213) (312) (411)
(123) (222) (321) (1113)
(132) (231) (1122) (2112)
(141) (1131) (1212) (3111)
(1221) (2121) (11112)
(1311) (2211) (11121)
(11211) (21111)
(12111)
(111111)
MATHEMATICA
skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]), {i, Length[f]}];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], skats[#]==k&]], {n, 0, 10}, {k, -n, n, 2}]
CROSSREFS
The central column k=0 is A001700 (aerated), half A357641.
Row sums are A011782.
For original alternating sum we have A097805, unordered A344651.
The skew-alternating sum of standard compositions is A357623, half A357621.
The case of partitions is A357638, half A357637.
The half-alternating version is A357645.
The reverse version for partitions is A357705, half A357704.
Sequence in context: A088234 A228717 A215977 * A185813 A300756 A347062
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Oct 12 2022
STATUS
approved