Wolfram Language (Mathematica), 28 bytes

\!\(\_{x,#}#2\)/#!/.x->0&

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Input [k, p], where p is a polynomial in terms of x. Outputs \$c_k\$.

 is \[DifferenceDelta]. This character has code point U+F4A4; the documentation incorrectly claims that it has code point U+2206. That character, ∆, is actually the similar-looking \[Laplacian] (undocumented).

Returns \$\frac{\Delta^k p(0)}{k!}\$, where \$\Delta^k p\$ is the \$k\$th "discrete derivative" of \$p\$ defined by \begin{align*} \Delta^0 p(x)&=p(x),\\ \Delta^{n+1}p(x)&=\Delta^n p(x+1)-\Delta^n p(x). \end{align*}

(This is analogous to obtaining the coefficients of the usual power series representation via differentiation, where the coefficient of the \$x^k\$ term is \$\frac{D^k p(0)}{k!}\$.)

Wolfram Language (Mathematica), 28 bytes

\!\(\_{x,#}#2\)/#!/.x->0&

Try it online!

Input [k, p], where p is a polynomial in terms of x. Outputs \$c_k\$.

 is \[DifferenceDelta]. This character has code point U+F4A4; the documentation incorrectly claims that it has code point U+2206. That character, ∆, is actually the similar-looking \[Laplacian] (undocumented).

Returns \$\frac{\Delta^k p(0)}{k!}\$, where \$\Delta^k p\$ is the \$k\$th "discrete derivative" of \$p\$ defined by \begin{align*} \Delta^0 p(x)&=p(x),\\ \Delta^{n+1}p(x)&=\Delta^n p(x+1)-\Delta^n p(x). \end{align*}

(This is analogous to obtaining the coefficients of the usual power series representation via differentiation, where the coefficient of the \$x^k\$ term is \$\frac{D^k p(0)}{k!}\$.)

Wolfram Language (Mathematica), 27 bytes

0~Range~#~StirlingS2~#2.#3&

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Input [n, k, p], where n is the degree of the polynomial and p is an ascending list of coefficients . Outputs \$c_k\$.

Wolfram Language (Mathematica), 28 bytes

\!\(\_{x,#}#2\)/#!/.x->0&

Try it online!

Input [k, p], where p is a polynomial in terms of x. Outputs the \$k\$th coefficient.

 is \[DifferenceDelta]. This character has code point U+F4A4; the documentation incorrectly claims that it has code point U+2206. That character, ∆, is actually the similar-looking \[Laplacian] (undocumented).

Returns \$\frac{\Delta^k p(0)}{k!}\$, where \$\Delta^k p\$ is the \$k\$th "discrete derivative" of \$p\$ defined by \begin{align*} \Delta^0 p(x)&=p(x),\\ \Delta^{n+1}p(x)&=\Delta^n p(x+1)-\Delta^n p(x). \end{align*}

Wolfram Language (Mathematica), 28 bytes

\!\(\_{x,#}#2\)/#!/.x->0&

Try it online!

Input [k, p], where p is a polynomial in terms of x. Outputs \$c_k\$.

 is \[DifferenceDelta]. This character has code point U+F4A4; the documentation incorrectly claims that it has code point U+2206. That character, ∆, is actually the similar-looking \[Laplacian] (undocumented).

Returns \$\frac{\Delta^k p(0)}{k!}\$, where \$\Delta^k p\$ is the \$k\$th "discrete derivative" of \$p\$ defined by \begin{align*} \Delta^0 p(x)&=p(x),\\ \Delta^{n+1}p(x)&=\Delta^n p(x+1)-\Delta^n p(x). \end{align*}

(This is analogous to obtaining the coefficients of the usual power series representation via differentiation, where the coefficient of the \$x^k\$ term is \$\frac{D^k p(0)}{k!}\$.)

Wolfram Language (Mathematica), 28 bytes

\!\(\_{x,#}#2\)/#!/.x->0&

Try it online!

Input [k, p], where p is a polynomial in terms of x. Outputs the \$k\$th coefficient.

 is \[DifferenceDelta]. This character has code point U+F4A4; the documentation incorrectly claims that it has code point U+2206. That character is actually the similar-looking ∆ (\[Laplacian], undocumented).

Wolfram Language (Mathematica), 28 bytes

\!\(\_{x,#}#2\)/#!/.x->0&

Try it online!

Input [k, p], where p is a polynomial in terms of x. Outputs the \$k\$th coefficient.

 is \[DifferenceDelta]. This character has code point U+F4A4; the documentation incorrectly claims that it has code point U+2206. That character, ∆, is actually the similar-looking \[Laplacian] (undocumented).

Returns \$\frac{\Delta^k p(0)}{k!}\$, where \$\Delta^k p\$ is the \$k\$th "discrete derivative" of \$p\$ defined by \begin{align*} \Delta^0 p(x)&=p(x),\\ \Delta^{n+1}p(x)&=\Delta^n p(x+1)-\Delta^n p(x). \end{align*}