Self Referential Polynomials
Mathematica, 55 bytes

NullSpace@Table[x^c-Boole[r==c]/.x-&gt;r,{r,0,#},{c,0,#}]&amp;


Output is the list coefficients, beginning from the constant term. Example:

In[1084] := Do[Print[%1077[n] // StandardForm], {n, 0, 7}]

{{1}}

{{0,1}}

{{-1,-1,1}}

{{19,-6,-29,10}}

{{-19,423,287,-325,57}}

{{-227362,-355855,44088,291323,-120862,12813}}

{{145991969,64989065,-123338281,-85635661,79841909,-18146731,1286795}}

{{-5958511844199,3384370785404,8437850634901,489428412300,-4499161007143,1776194531596,-258931801371,13131073916}}




This simply finds the vector such that (A - I)v = 0, similar to the MAPLE code in OEIS. The NullSpace method seems to always pick the minimal positive number for the last element, which matches the task description. 

The x^c-…/.x-&gt;r indirection is to prevent having 0^0 == Indeterminate.