Generating generating expressions for sequences
JavaScript (ES7),  193 ... 154  145 bytes
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Returns \$(a_0,a_1,...,a_k)\$ with some possible trailing zeros.

v=&gt;v.map((_,i)=&gt;(g=(i,m=v.map((n,y)=&gt;v.map((_,x)=&gt;x==i?n:y**x)))=&gt;+m||m.reduce((s,[v],i)=&gt;v*g(0,m.map(([,...r])=&gt;r).filter(_=&gt;i--))-s,0))(i)/g())

Try it online!
(removed the penultimate test case, which requires more precision than IEEE-754 provides)
How?
We use Cramer's rule to solve a system of linear equations based on a square Vandermonde matrix:

Given an input vector of length \$n\$, we build a Vandermonde matrix \$V_n\$ of size \$n\times n\$ with coefficients \$\alpha_i=i,0\le i &lt;n\$:
$$Vn=\begin{pmatrix}
1&amp;0&amp;0&amp;...&amp;0\\
1&amp;1&amp;1&amp;...&amp;1\\
1&amp;2&amp;4&amp;...&amp;2^{n-1}\\
\vdots&amp;\vdots&amp;\vdots&amp;\ddots&amp;\vdots\\
1&amp;n-1&amp;(n-1)^2&amp;...&amp;(n-1)^{n-1}
\end{pmatrix}$$

Using Cramer's rule, the coefficient \$a_i\$ of the polynomial is computed by taking the determinant of the matrix obtained by replacing the \$i\$-th column of \$V_n\$ with the input vector, and dividing by the determinant of \$V_n\$.


Example for \$(4,2,12,52,140)\$
The constant coefficient \$a_0\$ is given by:
$$a_0=\begin{vmatrix}
\color{blue}4&amp;0&amp;0&amp;0&amp;0\\
\color{blue}2&amp;1&amp;1&amp;1&amp;1\\
\color{blue}{12}&amp;2&amp;4&amp;8&amp;16\\
\color{blue}{52}&amp;3&amp;9&amp;27&amp;81\\
\color{blue}{140}&amp;4&amp;16&amp;64&amp;256
\end{vmatrix}/|V_5|=\frac{1152}{288}=4$$
The coefficient \$a_1\$ is given by:
$$a_1=\begin{vmatrix}
1&amp;\color{blue}4&amp;0&amp;0&amp;0\\
1&amp;\color{blue}2&amp;1&amp;1&amp;1\\
1&amp;\color{blue}{12}&amp;4&amp;8&amp;16\\
1&amp;\color{blue}{52}&amp;9&amp;27&amp;81\\
1&amp;\color{blue}{140}&amp;16&amp;64&amp;256
\end{vmatrix}/|V_5|=\frac{-576}{288}=-2$$
And so on.