additive combinatorics – How to prove the combinatorial identity $sum_{k=ell}^{n}binom{2n-k-1}{n-1}k2^k=2^ell nbinom{2n-ell}{n}$ for $ngeellge0$?

With the aid of the simple identity
begin{equation*}
sum_{k=0}^{n}binom{n+k}{k}frac{1}{2^{k}}=2^n
end{equation*}

in Item (1.79) on page 35 of the monograph

R. Sprugnoli, Riordan Array Proofs of Identities in Gould’s Book, University of Florence, Italy, 2006. (Has this monograph been formally published somewhere?)

I proved the combinatorial identity
$$
sum_{k=1}^{n}binom{2n-k-1}{n-1}k2^k=nbinom{2n}{n}, quad ninmathbb{N}.
$$

My question is: how to prove the more general combinatorial identity
$$
sum_{k=ell}^{n}binom{2n-k-1}{n-1}k2^k=2^ell nbinom{2n-ell}{n}
$$

for $ngeellge0$?