$\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{\displaystyle{#1}}\,} \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack} \newcommand{\dd}{\mathrm{d}} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,} \newcommand{\ic}{\mathrm{i}} \newcommand{\mc}[1]{\mathcal{#1}} \newcommand{\mrm}[1]{\mathrm{#1}} \newcommand{\pars}[1]{\left(\,{#1}\,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,} \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}} \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$
Note that only terms with $\ds{m \leq i \leq {n - 1 \over 2}}$ make a contribution to the sum. Also, it yields $\ds{n \geq 2m + 1}$.
\begin{align}\sum_{i = 0}^{n}{n \choose 2i + 1}{i \choose m} & =\sum_{k = m}^{\infty}{n \choose n - 2k - 1}{k \choose k - m} =\sum_{k = m}^{\infty}{n \choose n - 2k - 1}{-m - 1 \choose k - m}\pars{-1}^{k - m}\\[5mm] & =\sum_{k = 0}^{\infty}{n \choose n - 2m - 1 - 2k}{-m - 1 \choose k}\pars{-1}^{k}\\[5mm] & =\sum_{k = 0}^{\infty}{-m - 1 \choose k}\pars{-1}^{k}\oint_{\verts{z} = 1^{-}}{\pars{1 + z}^{n} \over z^{n - 2m - 2k}}\,{\dd z \over 2\pi\ic}\\[5mm] & =\oint_{\verts{z} = 1^{-}}{\pars{1 + z}^{n} \over z^{n - 2m}}\sum_{k = 0}^{\infty}{-m - 1 \choose k}\pars{-z^{2}}^{k}\,{\dd z \over 2\pi\ic} \\[5mm] & =\oint_{\verts{z} = 1^{-}}{\pars{1 + z}^{n} \over z^{n - 2m}}\pars{1 - z^{2}}^{-m - 1}\,{\dd z \over 2\pi\ic} =\oint_{\verts{z} = 1^{-}}{\pars{1 + z}^{n - m - 1} \over z^{n - 2m}\pars{1 - z}^{m + 1}}\,{\dd z \over 2\pi\ic}\\[5mm] & \stackrel{z\ \mapsto\ 1/z}{=}\,\,\,\oint_{\verts{z} = 1^{+}}{\pars{1 + z}^{n - m - 1} \over \pars{z - 1}^{m + 1}}\,{\dd z \over 2\pi\ic}\\[5mm] & =\oint_{\verts{z} = 1^{+}}{2^{n - m - 1}\bracks{1 + \pars{z - 1}/2}^{n - m - 1} \over\pars{z - 1}^{m + 1}}\,{\dd z \over 2\pi\ic}\\[5mm] & =2^{n - m - 1}\sum_{k = 0}^{n - m - 1}{n - m - 1 \choose{k}}{1 \over 2^{k}}\\underbrace{\oint_{\verts{z} = 1^{+}}{1 \over \pars{z - 1}^{m + 1 - k}}\,{\dd z \over 2\pi\ic}}_{\ds{\delta_{km}}}\\[5mm] & =\bbx{\ds{2^{n - 2m - 1}{n - m - 1 \choose m}}}\end{align}