The classical Euler's gamma function $\Gamma(z)$ can be defined by
\begin{equation}
\Gamma(z)=\lim_{n\to\infty}\frac{n!n^z}{\prod_{k=0}^n(z+k)}, \quad z\in\mathbb{C}\setminus\{0,-1,-2,\dotsc\}.
\end{equation}
The extended binomial coefficient $\binom{z}{w}$ for $z,w\in\mathbb{C}$ is defined by
\begin{equation}
\binom{z}{w}=
\begin{cases}
\dfrac{\Gamma(z+1)}{\Gamma(w+1)\Gamma(z-w+1)}, & z\not\in\mathbb{N}_-,\quad w,z-w\not\in\mathbb{N}_-;\\
0, & z\not\in\mathbb{N}_-,\quad w\in\mathbb{N}_- \text{ or } z-w\in\mathbb{N}_-;\\
\dfrac{\langle z\rangle_w}{w!},& z\in\mathbb{N}_-, \quad w\in\mathbb{N}_0;\\
\dfrac{\langle z\rangle_{z-w}}{(z-w)!}, & z,w\in\mathbb{N}_-, \quad z-w\in\mathbb{N}_0;\\
0, & z,w\in\mathbb{N}_-, \quad z-w\in\mathbb{N}_-;\\
\infty, & z\in\mathbb{N}_-, \quad w\not\in\mathbb{Z},
\end{cases}
\end{equation}
where
\begin{align}
\mathbb{Z}&=\{0,\pm1,\pm2,\dotsc\}, & \mathbb{N}&=\{1,2,\dotsc\},\\
\mathbb{N}_0&=\{0,1,2,\dotsc\}, & \mathbb{N}_-&=\{-1,-2,\dotsc\}
\end{align}
and
\begin{align}
\langle\alpha\rangle_n&=\prod_{k=0}^{n-1}(\alpha-k)\\
&=
\begin{cases}
\alpha(\alpha-1)\dotsm(\alpha-n+1), & n\ge1\\
1, & n=0
\end{cases}
\end{align}
is called the falling factorial.

By the idea of descending induction from Darij Grinberg, we can obtain the following more general identity.

Let $\ell,n\in\mathbb{Z}$ such that $\ell\le n$. Then
\begin{equation}
\sum_{k=\ell}^{n}\binom{2n-k-1}{n-1}2^kk
=\binom{2n-\ell}{n}2^\ell n.
\end{equation}
If an empty sum is understood to be $0$, then this identity holds for all integers $\ell,n\in\mathbb{Z}$, without the restriction $\ell\le n$.

`Sum[Binomial[2 n - k - 1, n - 1]*k*2^k, {k, s, n}, Assumptions -> n >= s && s >= 0 && {s, n} \[Element] Integers]`

results in $-2^s (s-2 n) \binom{2 n-s-1}{n-1}$, confirming the identity. $\endgroup$1more comment