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A357645
Triangle read by rows where T(n,k) is the number of integer compositions of n with half-alternating sum k, where k ranges from -n to n in steps of 2.
9
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, 0, 0, 9, 23, 80, 164, 230, 230, 168, 100, 9, 11
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
Row sums are A011782.
For original alternating sum we have A097805, unordered A344651.
Column k = n-4 appears to be A177787.
The case of partitions is A357637, skew A357638.
The central column k=0 is A357641 (aerated).
The skew-alternating version is A357646.
The reverse version for partitions is A357704, skew A357705.
A351005 = alternately equal and unequal partitions, compositions A357643.
A351006 = alternately unequal and equal partitions, compositions A357644.
A357621 gives half-alternating sum of standard compositions, skew A357623.
A357629 gives half-alternating sum of prime indices, skew A357630.
A357633 gives half-alternating sum of Heinz partition, skew A357634.
Sequence in context: A238727 A056885 A029373 * A366370 A297617 A351982
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Oct 12 2022
STATUS
approved