A261880 Array of higher-order differences of the sequence (-1)^n*A000111(n) read by downward antidiagonals.

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

Author

Paul Curtz, Jul 10 2016

Keywords

• sign
• tabl

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

Examples

			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

Cf. A000111, A000667, A062162, A227862, A239005.

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

Extensions

Edited by Wolfdieter Lang, Aug 08 2016