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A261880
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Array of higher-order differences of the sequence (-1)^n*A000111(n) read by downward antidiagonals.
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0
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1, -1, -2, 1, 2, 4, -2, -3, -5, -9, 5, 7, 10, 15, 24, -16, -21, -28, -38, -53, -77, 61, 77, 98, 126, 164, 217, 294, -272, -333, -410, -508, -634, -798, -1015, -1309, 1385, 1657, 1990, 2400, 2908, 3542, 4340, 5355, 6664
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OFFSET
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0,3
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COMMENTS
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Difference array of (-1)^n*A000111(n):
1, -1, 1, -2, 5, ...
-2, 2, -3, 7,...
4, -5, 10, ...
-9, 15, ...
24, ... .
Antidiagonal sums: b(n) = 1, -3, 7, -19, 61, -233, 1037, -5279, 30241, ..., i.e., row sums of the triangle.
Any triangle with entries T(n, m) built from some sequence in column m=0, and the recurrence T(n, m) = T(n, m-1) - T(n-1, m-1) for m >= 1, has the property that the new triangle t(n, m) = T(n+1, m+1) - T(n+1, m), 0 <= m <= n, equals -T(n, m). See the question in the example. - Wolfdieter Lang, Aug 08 2016
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LINKS
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FORMULA
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Recurrence: T(n, 0) = (-1)^n*A000111(n), n >= 0. T(n, m) = T(n, m-1) - T(n-1, m-1), m >= 1. (from the fact that the differences of the rows, starting with n = 1 produce the negative of the triangle. See the example and a comment). - Wolfdieter Lang, Aug 08 2016
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EXAMPLE
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The triangle T(n, m) begins:
n\m 0 1 2 3 4 5 ...
0: 1
1: -1 -2
2: 1 2 4
3: -2 -3 -5 -9
4: 5 7 10 15 24,
5: -16 -21 -28 -38 -53 -77
...
Triangle of differences of the row entries of the preceding triangle starting with row n=1:
n\m 0 1 2 3 4 ...
0: -1
1: 1 2
2: -1 -2 -4
3: 2 3 5 9
4: -5 -7 -10 -15 -24
... .
This is the negative of the first triangle. Are there other sequences with the same property?
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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