# 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 $$ngeellge0$$?