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

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 $ngeellge0$?