**Could you please recommend a reference to or supply a proof of the following identity \eqref{combin-ID-Maclaurin}, or \eqref{first-equiv-form}, or \eqref{combin-ID-Mac-Equiv}, or \eqref{combin-ID-Mac-Reform} for $m\ge2$?**

For non-negative integers $k,n\ge0$, I guess that \begin{equation}\label{combin-ID-Maclaurin}\tag{Q1} \sum_{\ell=0}^{n}\binom{2n+1}{2\ell+1} \binom{n-\ell}{n-k} =2^{2k}\frac{2n+1}{2k+1}\binom{n+k}{2k}. \end{equation}

The identity \eqref{combin-ID-Maclaurin} is equivalent to \begin{equation}\label{first-equiv-form}\tag{Q2} \sum_{\ell=0}^{k} \binom{2n+1}{2k-2\ell+1} \binom{\ell+n-k}{\ell} =2^{2k}\frac{2n+1}{2k+1}\binom{n+k}{2k}, \quad k,n\ge0 \end{equation} and is equivalent to \begin{equation}\label{combin-ID-Mac-Equiv}\tag{Q3} \sum_{\ell=0}^{n}\binom{2n+1}{2\ell+1} \binom{n-\ell}{m} =2^{2(n-m)}\frac{2n+1}{2(n-m)+1}\binom{2n-m}{m}, \quad n\ge0,\quad n\ge m\in\mathbb{Z}. \end{equation} The identity \eqref{combin-ID-Mac-Equiv} can be further reformulated as \begin{equation}\label{combin-ID-Mac-Reform}\tag{Q4} \sum_{\ell=0}^{n}\binom{2n+1}{2\ell} \binom{\ell}{m} =2^{2(n-m)}\frac{2n+1}{2(n-m)+1}\binom{2n-m}{m},\quad n\ge0,\quad n\ge m\in\mathbb{Z}. \end{equation}

When $m<0$, it is trivial that both sides of \eqref{combin-ID-Mac-Reform} are equal to $0$.

When $m=0$, the identity \eqref{combin-ID-Mac-Equiv} or \eqref{combin-ID-Mac-Reform} becomes \begin{equation}\label{m=0-(1.93)Sprug}\tag{Q5} \sum_{\ell=0}^{n}\binom{2n+1}{2\ell} =\sum_{\ell=0}^{n}\binom{2n+1}{2\ell+1} =2^{2n}, \quad n\ge0. \end{equation} This is just the identity (1.93) on page 38 in the monograph [1] below.

When $m=1$, the identity \eqref{combin-ID-Mac-Equiv} or \eqref{combin-ID-Mac-Reform} reduces to \begin{equation}\tag{Q6} \sum_{\ell=0}^{n}(n-\ell)\binom{2n+1}{2\ell+1} =\sum_{\ell=0}^{n}\binom{2n+1}{2\ell}\ell =2^{2(n-1)}(2n+1), \quad n\ge1. \end{equation} This follows from combining \eqref{m=0-(1.93)Sprug} with the identity \begin{equation}\tag{Q7} \sum_{\ell=1}^{n}\binom{2n+1}{2\ell+1}\ell=(2n-1)4^{n-1}, \quad n\ge1, \end{equation} which can be found in (1.100) on page 40 of the monograph [1] below.

**Could you please recommend a reference to or provide a proof of the above identity \eqref{combin-ID-Maclaurin}, or \eqref{first-equiv-form}, or \eqref{combin-ID-Mac-Equiv}, or \eqref{combin-ID-Mac-Reform} for $m\ge2$?**

References

- R. Sprugnoli,
*Riordan Array Proofs of Identities in Gould’s Book*, University of Florence, Italy, 2006. - F. Qi,
*Diagonal recurrence relations for the Stirling numbers of the first kind*, Contrib. Discrete Math.**11**(2016), no. 1, 22--30; available online at https://doi.org/10.11575/cdm.v11i1.62389. - F. Qi,
*Diagonal recurrence relations, inequalities, and monotonicity related to the Stirling numbers of the second kind*, Math. Inequal. Appl.**19**(2016), no. 1, 313--323; available online at https://doi.org/10.7153/mia-19-23. - F. Qi and B.-N. Guo,
*A diagonal recurrence relation for the Stirling numbers of the first kind*, Appl. Anal. Discrete Math.**12**(2018), no. 1, 153--165; available online at https://doi.org/10.2298/AADM170405004Q.