[seqfan] new array with polynomial coefficients A197184

Richard Mathar mathar at strw.leidenuniv.nl
Tue Oct 11 17:46:19 CEST 2011

```I have stumbled across an array of polynomial coefficients

1

-1 + x

2
-1 - x + x

2    3
7 - 2 x - x  + x

2    3    4
-13 + 12 x - 3 x  - x  + x

2      3    4    5
-17 - 22 x + 18 x  - 4 x  - x  + x

2       3      4    5    6
199 - 45 x - 35 x  + 25 x  - 5 x  - x  + x

2       3       4      5    6    7
-605 + 465 x - 84 x  - 53 x  + 33 x  - 6 x  - x  + x

2        3       4       5      6    7    8
-225 - 1449 x + 910 x  - 133 x  - 77 x  + 42 x  - 7 x  - x  + x

2         3        4        5       6      7    8    9
11703 - 864 x - 3094 x  + 1594 x  - 190 x  - 108 x  + 52 x  - 8 x  - x  + x

1,
-1,1,
-1,-1,1,
7,-2,-1,1,
-13,12,-3,-1,1,
-17,-22,18,-4,-1,1,
199,-45,-35,25,-5,-1,1,
-605,465,-84,-53,33,-6,-1,1,
-225,-1449,910,-133,-77,42,-7,-1,1,
11703,-864,-3094,1594,-190,-108,52,-8,-1,1,
-59317,33780,-1380,-6027,2583,-252,-147,63,-9,-1,1,
83143,-179398,78567,-771,-10899,3948,-315,-195,75,-10,-1,1,
991671,271073,-461978,159115,2882,-18546,5764,-374,-253,88,-11,-1,1,
-9079829,3285597,882801,-1054204,291516,12774,-30009,8109,-423,-322,102,-12,-1,1,
34001447,-32206525,8244457,2558153,-2188667,494091,33787,-46553,11063,-455,-403,117,-13,-1,1,
56463335,131329838,-93561979,17151067,6552040,-4211647,785917,72952,-69685,14707,-462,-497,133,-14,-1,1,
-1748822285,168315368,439601283,-235475361,30544802,15107811,-7614447,1184578,139961,-101171,19122,-435,-605,150,-15,-1,1,

defined in http://oeis.org/A074051 and http://oeis.org/A074052 and cemented in http://oeis.org/A197184.
For those who like Stirling and Bernoulli-related arrays, it may be some
sort of challenge to figure out what the recurrences, generating functions
etc might be. Apparently one gets polynomial recurrences down each column
if the rows are reversed, i.e., if the polynomial coefficients are listed
along declining exponents.

Richard Mathar

```

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