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A357638
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Triangle read by rows where T(n,k) is the number of integer partitions of n with skew-alternating sum k, where k ranges from -n to n in steps of 2.
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24
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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
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OFFSET
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0,13
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COMMENTS
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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 + ....
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LINKS
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FORMULA
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Conjecture: The columns are palindromes with sums A298311.
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EXAMPLE
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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)
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MATHEMATICA
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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}]
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CROSSREFS
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Number of nonzero entries in row n appears to be A004396(n+1).
First nonzero entry of each row appears to converge to A146325.
The half-alternating version is A357637.
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.
Cf. A035594, A053251, A357136, A357189, A357486, A357487, A357488, A357624, A357631, A357632, A357636.
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KEYWORD
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AUTHOR
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STATUS
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approved
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