[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
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