#JavaScript (ES7), <s>&nbsp;193 181 160&nbsp;</s> 159 bytes

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

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

[Try it online!](https://tio.run/##nZXRbqMwEEXf@xU8rezUENsYA62gL9nH/YIIVSghKVWAiqQolfrv2RkbUlRttZAoBAdyr@@csclr3uXHTVu@ndy62RaXXXLpkrTzqvyNkGdW0iQlq6RK0vvq87Py2mL7vikIObJ1l5m7x3tS/hJPbvfQ0cWKVFa5Zp7ntRncb6m3Kw@noiXPSVq6LqWMU0rIPimv89Tsg44mPcOXc5KUT/XDx2JxpvDzktLliuwJjC@P6zvHWbsyY86813JpZKjmzOEz9ajmRiyZg@85ehTbmRVz1A0zKyN2/dm5TdEg84e6ITpk0JNdTN1Ybl@6viE9zmn0ECS8Qe8PekgeMUeAn5iaw9Ab9EjCFcwJwAAOGf/fZAB4ZQADYRjGcxmicsAAJcCnhitikk2PwZUjD8ggsLXgGwQTXAaPMU5kKYFpAOdITcOhzNRqWBG9iw9HyCfGkAbiGAigkNgZNaOvfTf9wUYZrhgGyxGKT64HuX5tE4HNhiPGczRpx6GPsD3GquyTAl2BbYR0TCIuEbev4aIK/egfxvYhZaC6dpn5Q2mBZSx8zDkh1ag0u@8tJMs9hJNWCD0KQ4gYxziWOgx@CCVsIG458VEufCiYJYgOgYJhxGO8znUsf3AbkPeO8kre2LkiRFgIMsBhGGuTlEv8EglcsCLSUmPYPly/OZTN97VPLDiNixPBKaGNbaRj7IHEFgsB5YfZ94D96kK761rN7rxd0/7ONy8kd5LU2TT1sTkU3qHZkx3JqfkP6/DOn/z04rXNe70lHcX/rb8 "JavaScript (Node.js) – Try It Online")

(the above test link filters out floating point approximation errors by rounding the results; this is actually only needed for the penultimate test case)

###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 computing 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:

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


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