#JavaScript (ES7), <s>&nbsp;193 ... 156&nbsp;</s> 154 bytes

Returns \$(a_0,a_1,...,a_k)\$ with some possible trailing zeros.

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    v=>v.map((_,i)=>(g=(i,m=v.map((n,y)=>v.map((_,x)=>x==i?n:y**x)))=>+m||m.reduce((s,[v],i)=>s+(i&1?-v:v)*g(0,m.map(([,...r])=>r).filter(_=>i--)),0))(i)/g())

[Try it online!](https://tio.run/##nZXfbpswFMbv@xS@muzUJrYx/zqRXu0pIlShlGRMIVSkQ6nUd8/OsXHHplUziUIAi@/zd37HOD/qsT7vhvblVZz65@a6L69juRmjrn6h9Im3rNzQQ0lb3pXT4Im/sdkTF7i5lGX7eHp4W60ujMH9fff@3kVD8/xz11B65tuxsk7ne9p@UY9ifBjZ6kAl75zLlkdRNFTwxMCifXt8bQb6VG5aIRjjkjHasvWBMnb9ur0jZCt0xcmyz3ptZaiWnMiFelRLK9ac4HeJHsVuZsOJuWFmY8UiXpzbFg2y2NcN0SFDGuxi68Zyp9LTG9LjnFYPQbIb9LHXQ/KcEwV@KjSHpef1SEIoThIwgEMX/zfxAD8YwIWyDIulDFHpMUAJ8JvCiAqymTAIPfOADApbC75JEuDiPeY4kaUGpgmccxOGw9ipjV8Rk0sMRyYDY2gLcQ4EUGjsjFnQ16mbsbcxliuGwXKUkcH1INffr4nCZsNR4DkPeuPQR7keY1Vup0BXYJsjHZtIasQdpzBosjj/h7HbpCxU4ZZZ7EtLHGMVY86AVLPS3HvvIDnuGZxSg9DzLIOIRYHXOs2ST0IpF0g6TnKWCzcFuwTRITFwmcsCx2Va6E/cPPLJUf9ZZopLCcs0Cp1VkqcFEtPYEKUgbFb9bTetBQz3sbKqu2jfD9/q3Xdak3JDdv3p3B@b6Ngf6J7W8D91/QU "JavaScript (Node.js) – Try It Online")

(removed the penultimate test case, which requires more precision than IEEE-754 provides)

###How?

We use [Cramer's rule][1] to solve a system of linear equations based on a square [Vandermonde matrix][2].

The Vandermonde matrix of size \$n\times n\$ that we are using is defined as:

$$Vn=\begin{pmatrix}
1&0&0&...&0\\
1&1&1&...&1\\
1&2&4&...&2^{n-1}\\
\vdots&\vdots&\vdots&&\vdots\\
1&n-1&(n-1)^2&...&(n-1)^{n-1}
\end{pmatrix}$$

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:

$$a_0=\begin{vmatrix}
\color{blue}4&0&0&0&0\\
\color{blue}2&1&1&1&1\\
\color{blue}{12}&2&4&8&16\\
\color{blue}{52}&3&9&27&81\\
\color{blue}{140}&4&16&64&256
\end{vmatrix}/|V_5|=\frac{1152}{288}=4$$

The coefficient \$a_1\$ is given by:

$$a_1=\begin{vmatrix}
1&\color{blue}4&0&0&0\\
1&\color{blue}2&1&1&1\\
1&\color{blue}{12}&4&8&16\\
1&\color{blue}{52}&9&27&81\\
1&\color{blue}{140}&16&64&256
\end{vmatrix}/|V_5|=\frac{-576}{288}=-2$$

And so on.

  [1]: https://en.wikipedia.org/wiki/Cramer%27s_rule
  [2]: https://en.wikipedia.org/wiki/Vandermonde_matrix