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
$$
\sum\_{k=1}^{n}\binom{2n-k-1}{n-1}k2^k=n\binom{2n}{n}, \quad n\in\mathbb{N}.
$$


My question is: how to prove the more general combinatorial identity
$$
\sum\_{k=\ell}^{n}\binom{2n-k-1}{n-1}k2^k=2^\ell n\binom{2n-\ell}{n}
$$
for $n\ge\ell\ge0$?



How to prove the combinatorial identity $\sum_{k=\ell}^{n}\binom{2n-k-1}{n-1}k2^k=2^\ell n\binom{2n-\ell}{n}$ for $n\ge\ell\ge0$?
For $k\in\mathbb{N}$, let $s\_k$ and $S\_k$ be two sequences independent of $n$ such that $n\ge k\in\mathbb{N}$. The inversion theorem, Theorem 4.4 on page 528 in the freely downloading paper [1] below, reads that
\begin{equation}\label{Qi-Zou-Guo-Inversion-thm}\tag{1}
s\_n=\sum\_{k=1}^{n}\binom{k}{n-k}S\_k
\quad\text{if and only if}\quad
(-1)^nnS\_n=\sum\_{k=1}^{n}\binom{2n-k-1}{n-1}(-1)^kks\_k.
\end{equation}
Applying the inversion theorem in \eqref{Qi-Zou-Guo-Inversion-thm} and considering the identity
\begin{equation\*}
\sum\_{k=1}^{n}\binom{2n-k-1}{n-1}2^kk
=2n\binom{2n-1}{n},
\end{equation\*}
which can be deduced from letting $\ell=1$ in the resulted identity of the above answers, we conclude
\begin{equation\*}
\sum\_{k=1}^{n}(-1)^k\binom{k}{n-k}\binom{2k-1}{k}=(-1)^n2^{n-1}, \quad n\in\mathbb{N}.
\end{equation\*}
See Remark 3.4 on page 11 in the paper [2] below.


References


1. Feng Qi, Qing Zou, and Bai-Ni Guo, *The inverse of a triangular matrix and several identities of the Catalan numbers*, Applicable Analysis and Discrete Mathematics **13** (2019), no. 2, 518--541; available online at <https://doi.org/10.2298/AADM190118018Q>.
2. Feng Qi and Mark Daniel Ward, *Closed-form formulas and properties of coefficients in Maclaurin's series expansion of Wilf's function*, arXiv (2021), available online at <https://arxiv.org/abs/2110.08576v1>.


