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A261880
Array of higher-order differences of the sequence (-1)^n*A000111(n) read by downward antidiagonals.
0
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
OFFSET
0,3
COMMENTS
Difference array of (-1)^n*A000111(n):
1, -1, 1, -2, 5, ...
-2, 2, -3, 7,...
4, -5, 10, ...
-9, 15, ...
24, ... .
First column:(-1)^n*A000667(n).
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
FORMULA
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
EXAMPLE
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?
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Paul Curtz, Jul 10 2016
EXTENSIONS
Edited by Wolfdieter Lang, Aug 08 2016
STATUS
approved