Definition 1. Let $f_n\!\left(z\right)\coloneqq\left(1-z\right)^{n+1}\sum_{k=1}^\infty k^nz^k,$ where $n$ is a positive integer.

Our goal is to prove that $f_n\!\left(z\right)$ is a polynomial of degree $n$ w.r.t. $z$, and the sum of its coefficients is $n!$.

Lemma 1. $f_{n+1}\!\left(z\right)=z\,\left(1-z\right)^{n+2}\frac{\mathrm d}{\mathrm dz}\left(\frac{f_n\!\left(z\right)}{\left(1-z\right)^{n+1}}\right).$

Proof. \begin{align*} \frac{\mathrm d}{\mathrm dz}\left(\frac{f_n\!\left(z\right)}{\left(1-z\right)^{n+1}}\right) &=\frac{\mathrm d}{\mathrm dz}\sum_{k=1}^\infty k^nz^k\\ &=\sum_{k=1}^\infty k^{n+1}z^{k-1}\\ &=\frac{f_{n+1}\!\left(z\right)}{z\,\left(1-z\right)^{n+2}}. \end{align*} $\square$

Definition 2 (Eulerian numbers). $\left<\begin{matrix}n\\k\end{matrix}\right>\coloneqq\sum_{j=0}^{k+1}\left(-1\right)^j\binom{n+1}j\left(k-j+1\right)^n.$

Lemma 2. $\left<\begin{matrix}n+1\\k+1\end{matrix}\right>=\left(n-k\right)\left<\begin{matrix}n\\k\end{matrix}\right>+\left(k+2\right)\left<\begin{matrix}n\\k+1\end{matrix}\right>.$

Proof. \begin{align*} &\phantom{=}~\,\left(n-k\right)\left<\begin{matrix}n\\k\end{matrix}\right>+\left(k+2\right)\left<\begin{matrix}n\\k+1\end{matrix}\right>\\ &=\left(n-k\right)\sum_{j=0}^{k+1}\left(-1\right)^j\binom{n+1}j\left(k-j+1\right)^n+\left(k+2\right)\sum_{j=0}^{k+2}\left(-1\right)^j\binom{n+1}j\left(k-j+2\right)^n\\ &=\left(n-k\right)\sum_{j=0}^{k+2}\left(-1\right)^{j-1}\binom{n+1}{j-1}\left(k-j+2\right)^n+\left(k+2\right)\sum_{j=0}^{k+2}\left(-1\right)^j\binom{n+1}j\left(k-j+2\right)^n\\ &=\sum_{j=0}^{k+2}\left(-1\right)^j\left(k-j+2\right)^n\left(\left(k-n\right)\binom{n+1}{j-1}+\left(k+2\right)\binom{n+1}j\right)\\ &=\sum_{j=0}^{k+2}\left(-1\right)^j\left(k-j+2\right)^n\left(\left(k-n\right)\frac{\left(n+1\right)!}{\left(j-1\right)!\left(n-j+2\right)!}+\left(k+2\right)\frac{\left(n+1\right)!}{j!\left(n-j+1\right)!}\right)\\ &=\sum_{j=0}^{k+2}\left(-1\right)^j\left(k-j+2\right)^n\frac{\left(n+1\right)!}{j!\left(n-j+2\right)}\left(\left(k-n\right)j+\left(k+2\right)\left(n-j+2\right)\right)\\ &=\sum_{j=0}^{k+2}\left(-1\right)^j\left(k-j+2\right)^n\frac{\left(n+1\right)!}{j!\left(n-j+2\right)}\left(n+2\right)\left(k-j+2\right)\\ &=\sum_{j=0}^{k+2}\left(-1\right)^j\left(k-j+2\right)^{n+1}\frac{\left(n+2\right)!}{j!\left(n-j+2\right)}\\ &=\sum_{j=0}^{k+2}\left(-1\right)^j\binom{n+2}j\left(k-j+2\right)^{n+1}\\ &=\left<\begin{matrix}n+1\\k+1\end{matrix}\right>. \end{align*} $\square$

Lemma 3. $\left<\begin{matrix}n\\0\end{matrix}\right>=1,\quad\left<\begin{matrix}n\\n\end{matrix}\right>=0.$

Brief proof. Easily proved by Definition 2. $\square$

Lemma 4. $f_n\!\left(z\right)=\sum_{k=1}^n\left<\begin{matrix}n\\n-k\end{matrix}\right>z^k.$

Proof. By mathematical induction. When $n=1$, $f_n\left(z\right)=z=\sum_{k=1}^n\left<\begin{matrix}n\\n-k\end{matrix}\right>z^k,$ the result holds.

Suppose the result holds when $n=n_0$, and then when $n=n_0+1$, \begin{align*} &\phantom{=}~\,f_n\!\left(z\right)\\ &=z\,\left(1-z\right)^{n_0+2}\frac{\mathrm d}{\mathrm dz}\left(\frac{f_{n_0}\left(z\right)}{\left(1-z\right)^{n_0+1}}\right) &\text{(Lemma 1)}\\ &=z\,\left(1-z\right)^{n_0+2}\frac{\frac{\mathrm df_{n_0}\left(z\right)}{\mathrm dz}\left(1-z\right)^{n_0+1}-f_{n_0}\!\left(z\right)\frac{\mathrm d\left(\left(1-z\right)^{n_0+1}\right)}{\mathrm dz}}{\left(1-z\right)^{2n_0+2}}\\ &=z\,\left(1-z\right)^{n_0+2}\frac{\frac{\mathrm df_{n_0}\left(z\right)}{\mathrm dz}\left(1-z\right)^{n_0+1}+\left(n_0+1\right)f_{n_0}\!\left(z\right)\left(1-z\right)^{n_0}}{\left(1-z\right)^{2n_0+2}}\\ &=z\left(\frac{\mathrm df_{n_0}\!\left(z\right)}{\mathrm dz}\left(1-z\right)+\left(n_0+1\right)f_{n_0}\left(z\right)\right)\\ &=z\left(\left(1-z\right)\frac{\mathrm d}{\mathrm dz}\sum_{k=1}^{n_0}\left<\begin{matrix}n_0\\n_0-k\end{matrix}\right>z^k+\left(n_0+1\right)\sum_{k=1}^{n_0}\left<\begin{matrix}n_0\\n_0-k\end{matrix}\right>z^k\right) &\text{(supposed)}\\ &=z\left(\left(1-z\right)\sum_{k=1}^{n_0}\left<\begin{matrix}n_0\\n_0-k\end{matrix}\right>kz^{k-1}+\left(n_0+1\right)\sum_{k=1}^{n_0}\left<\begin{matrix}n_0\\n_0-k\end{matrix}\right>z^k\right)\\ &=z\left(\sum_{k=1}^{n_0}\left<\begin{matrix}n_0\\n_0-k\end{matrix}\right>kz^{k-1}-\sum_{k=1}^{n_0}\left<\begin{matrix}n_0\\n_0-k\end{matrix}\right>kz^k+\left(n_0+1\right)\sum_{k=1}^{n_0}\left<\begin{matrix}n_0\\n_0-k\end{matrix}\right>z^k\right)\\ &=z\left(\sum_{k=0}^{n_0}\left<\begin{matrix}n_0\\n_0-k-1\end{matrix}\right>\left(k+1\right)z^k-\sum_{k=0}^{n_0}\left<\begin{matrix}n_0\\n_0-k\end{matrix}\right>kz^k+\left(n_0+1\right)\sum_{k=0}^{n_0}\left<\begin{matrix}n_0\\n_0-k\end{matrix}\right>z^k\right) &\text{(Lemma 3)}\\ &=z\sum_{k=0}^{n_0}\left(\left<\begin{matrix}n_0\\n_0-k-1\end{matrix}\right>\left(k+1\right)z^k-\left<\begin{matrix}n_0\\n_0-k\end{matrix}\right>kz^k+\left(n_0+1\right)\left<\begin{matrix}n_0\\n_0-k\end{matrix}\right>z^k\right)\\ &=\sum_{k=0}^{n_0}\left(\left<\begin{matrix}n_0\\n_0-k-1\end{matrix}\right>\left(k+1\right)-\left<\begin{matrix}n_0\\n_0-k\end{matrix}\right>k+\left(n_0+1\right)\left<\begin{matrix}n_0\\n_0-k\end{matrix}\right>\right)z^{k+1}\\ &=\sum_{k=1}^{n_0+1}\left(k\left<\begin{matrix}n_0\\n_0-k\end{matrix}\right>+\left(n_0-k+2\right)\left<\begin{matrix}n_0\\n_0-k+1\end{matrix}\right>\right)z^k\\ &=\sum_{k=1}^{n_0+1}\left<\begin{matrix}n_0+1\\n_0-k+1\end{matrix}\right>z^k &\text{(Lemma 2)}\\ &=\sum_{k=1}^n\left<\begin{matrix}n\\n-k\end{matrix}\right>z^k. \end{align*} Then we can derive that the result is true by mathematical induction. $\square$

Lemma 5. $\sum_{j=0}^n\left(-1\right)^{n-j}\binom njj^n=n!.$

Proof. Because $\mathrm e^x-1\sim x$ (in terms of infinitesimal quantity), $\left(\mathrm e^x-1\right)^n\sim x^n$, i.e. $\left(\mathrm e^x-1\right)^n=x^n+o\!\left(x^n\right)$ (where $o$ denotes higher order of infinitesimal quantity).

Apply Newton’s binomial theorem to the left-hand side, and we have $\sum_{j=0}^n\left(-1\right)^{n-j}\binom nj\mathrm e^{jx}=x^n+o\!\left(x^n\right).$ Take $n$th derivative of the equation, and we have $\sum_{j=0}^n\left(-1\right)^{n-j}\binom njj^n\mathrm e^{jx}=n!+o\!\left(1\right).$ Take $x=0$, and we have $\sum_{j=0}^n\left(-1\right)^{n-j}\binom njj^n=n!.$ $\square$

Lemma 6. $\sum_{j=0}^n\left(-1\right)^{n-j}\binom nj\left(j+1\right)^n=n!.$

Proof. \begin{align*} \left(n+1\right)!&=\sum_{j=0}^n\left(-1\right)^{n-j+1}\binom{n+1}jj^{n+1}& \text{(Lemma 5)}\\ &=\sum_{j=1}^n\left(-1\right)^{n-j+1}\binom{n+1}jj^{n+1}\\ &=\sum_{j=1}^n\left(-1\right)^{n-j+1}\frac{\left(n+1\right)!}{j!\left(n-j+1\right)!}j^{n+1} \\ &=\sum_{j=1}^n\left(-1\right)^{n-j+1}\frac{\left(n+1\right)n!}{\left(j-1\right)!\left(n-j+1\right)!}j^n\\ &=\sum_{j=0}^n\left(-1\right)^{n-j}\frac{\left(n+1\right)n!}{j!\left(n-j\right)!}\left(j+1\right)^n\\ &=\left(n+1\right)\sum_{j=0}^n\left(-1\right)^{n-j}\binom nj\left(j+1\right)^n. \end{align*} $\square$

Lemma 7. $\sum_{k=0}^n\left<\begin{matrix}n\\k\end{matrix}\right>=n!.$

Proof. \begin{align*} &\phantom{=}~\,\sum_{k=0}^n\left<\begin{matrix}n\\k\end{matrix}\right>\\ &=\sum_{k=0}^n\sum_{j=0}^{k+1}\left(-1\right)^j\binom{n+1}j\left(k-j+1\right)^n\\ &=\sum_{k=0}^n\sum_{j=0}^k\left(-1\right)^j\binom{n+1}j\left(k-j+1\right)^n\\ &=\sum_{j=0}^n\sum_{k=j}^n\left(-1\right)^j\binom{n+1}j\left(k-j+1\right)^n\\ &=\sum_{j=0}^n\left(-1\right)^j\binom{n+1}j\sum_{k=j}^n\left(k-j+1\right)^n\\ &=\sum_{j=0}^n\left(-1\right)^j\binom{n+1}j\sum_{k=1}^{n-j+1}k^n\\ &=\sum_{j=0}^n\left(-1\right)^{n-j}\binom{n+1}{n-j}\sum_{k=1}^{j+1}k^n\\ &=\sum_{j=0}^n\left(-1\right)^{n-j}\binom{n+1}{j+1}\sum_{k=1}^{j+1}k^n\\ &=\sum_{j=0}^n\left(-1\right)^{n-j}\left(\binom nj+\binom n{j+1}\right)\sum_{k=1}^{j+1}k^n\\ &=\sum_{j=0}^n\left(-1\right)^{n-j}\binom nj\sum_{k=1}^{j+1}k^n+\sum_{j=0}^n\left(-1\right)^{n-j}\binom n{j+1}\sum_{k=1}^{j+1}k^n\\ &=\sum_{j=0}^n\left(-1\right)^{n-j}\binom nj\left(j+1\right)^n+\sum_{j=0}^n\left(-1\right)^{n-j}\binom nj\sum_{k=1}^jk^n+\sum_{j=1}^n\left(-1\right)^{n-j+1}\binom nj\sum_{k=1}^jk^n\\ &=n!+\sum_{j=1}^n\left(-1\right)^{n-j}\binom nj\sum_{k=1}^jk^n-\sum_{j=1}^n\left(-1\right)^{n-j}\binom nj\sum_{k=1}^jk^n &\text{(Lemma 6)}\\ &=n!. \end{align*} $\square$

Proof of the original proposition. By Lemma 4, $f_n\!\left(z\right)$ is a polynomial of degree $n$ in $z$ (Lemma 3 guarantees that the coefficient of the $n$th degree term is not $0$).

The sum of coefficients \begin{align*} \sum_{k=1}^n\left<\begin{matrix}n\\n-k\end{matrix}\right> &=\sum_{k=0}^{n-1}\left<\begin{matrix}n\\k\end{matrix}\right>\\ &=\sum_{k=0}^n\left<\begin{matrix}n\\k\end{matrix}\right> &\text{(Lemma 3)}\\ &=n!. &\text{(Lemma 7)} \end{align*} $\square$