For every given degree \$n\$ it is possible to construct (at least one) an integral polynomial \$p \in \mathbb Z[X]\$ such that \$p(k)\$ (\$p\$ evaluated in \$k\$) is the coefficient of the term \$x^k\$ in the polynomial for all \$0 \leqslant k \leqslant n\$. To make them unique, we require the leading coefficient (the coefficient of \$x^n\$) to be positive and minimal.
One of the a priori unexpected properties is how the roots behave depending on \$n\$:
Given a nonnegative integer \$n\$ output the self referential integral polynomial of degree \$n\$ with minimal positive leading coefficient.
The output can be in any human readable form, as string
x^2-x-1, or also as a list of coefficients
[1,-1,-1]. (The order of the coefficients can also be the other way around, it just needs to be consistent.)
First few outputs
n=0: 1 n=1: x n=2: x^2-x-1 n=3: 10*x^3-29*x^2-6*x+19 n=4: 57*x^4-325*x^3+287*x^2+423*x-19 n=5: 12813*x^5-120862*x^4+291323*x^3+44088*x^2-355855*x-227362