# [seqfan] Divisibility sequences

Robert Munafo mrob27 at gmail.com
Wed Dec 16 17:02:44 CET 2009

```"rwg" wrote:
> This offers a good OEIS signature,
> but how to tabulate the actual, jagged, nontriangular coefficient array?

Nontriangular arrays can be listed by antidiagonals. A few random examples
include A052154, A114002, A114004, A161780.

So, assuming the polynomials are (I'm not sure, the exponents were hard to
line up with the x's in the email): 1, 1, -1, x + 1, x^2 - x - 1, -x^3 -x -
1, -3x^2 -2x, x^5 - x^4 -3x^2 +3x + 1

You arrange them with the x^i coefficient in the ith column and jth
polynomial in the jth row, so the "jagged triangle" becomes a regular array:

1  0  0  0  0  0  0 . . .
1  0  0  0  0  0  0
-1  0  0  0  0  0  0
1  1  0  0  0  0  0
-1 -1  1  0  0  0  0
-1 -1  0 -1  0  0  0
0 -2 -3  0  0  0  0
1  3 -3 -1  1  0  0 . . .
. . .
. . .

and the sequence is this array read by antidiagonals:

1 1 0 -1 0 0 1 0 0 0 -1 1 0 0 0 -1 -1 0 0 0 0 0 -1 1 0 0 0 0 1 -2 0 0 0 0 0
0 ...

If you have a 0th row for the 0th polynomial then there are more 0's:

1 0 1 0 0 -1 0 0 0 1 0 0 0 0 -1 1 0 0 0 0 -1 -1 0 0 0 0 0 0 -1 1 0 0 0 0 0
-1 -1 0 0 0 0 0 0 ...

--
Robert Munafo  --  mrob.com

```