Computing a sum involving factorials 0. Background. This question is linked to a previous one: https://math.stackexchange.com/questions/3950321/computing-sums-of-exponential-partial-bell-polynomials.
Based on the computation of the exponential partial Bell polynomial $B_{n,k}(2!,\ldots,(n-k+2)!)$ there (that I hope is correct), I managed to rewrite the problem and this led me to ask this new question.
1. The question. Can we explicitly compute
$$S(\alpha',\beta')=\sum_{\gamma=0}^{\min\{\beta',\,\alpha'-\beta'\}}\frac{2^{\beta'-\gamma}(-1)^\gamma}{\gamma!(\beta'-\gamma)!}\frac{(2\beta')^{\overline{\alpha'-\beta'-\gamma}}}{(\alpha'-\beta'-\gamma)!}$$
where $\beta'\geq1$ and $2\beta'\geq\alpha'\geq0$? Here $x^{\overline{k}}:=x(x+1)\ldots(x+k-1)$ denotes the rising factorial. If no closed form can be found, an estimate will be enough. Below are displayed two attempts to deal with the problem ; any advice to go further would be very appreciated.
2.1. First attempt. Introduce the signed Lah number (see https://en.wikipedia.org/wiki/Lah_number):
$$L(n,k):=(-1)^n\frac{n!(n-1)!}{k!(k-1)!(n-k)!}.$$
Then, putting $\gamma':=\alpha'-\beta'-\gamma$, we can write:
\begin{align*}
S(\alpha',\beta')&=\sum_{\gamma'=\min\{\alpha'-2\beta',\,0\}}^{\alpha'-\beta'}\frac{2^{2\beta'-\alpha'+\gamma'}(-1)^{\alpha'-\beta'-\gamma'}}{(\alpha'-\beta'-\gamma')!(2\beta'-\alpha'+\gamma')!}\frac{(2\beta')^{\overline{\gamma'}}}{\gamma'!}\\
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&=2^{2\beta'-\alpha'}\sum_{\gamma'=\min\{\alpha'-2\beta',\,0\}}^{\alpha'-\beta'}\frac{2^{\gamma'}(-1)^{\gamma'}}{(\alpha'-\beta'-\gamma')!(2\beta'-\alpha'+\gamma')!}\frac{(2\beta')^{\overline{\gamma'}}}{\gamma'!}\\
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&=\frac{2^{2\beta'-\alpha'}(-1)^{\alpha'-\beta'}}{(\alpha'-\beta')!(\alpha'-\beta'-1)!}\\&\quad\times\sum_{\gamma'=\min\{\alpha'-2\beta',\,0\}}^{\alpha'-\beta'}\frac{2^{\gamma'}(\gamma'-1)!}{(2\beta'-\alpha'+\gamma')!}(-1)^{\gamma'}\underbrace{(-1)^{\alpha'-\beta'}\frac{(\alpha'-\beta')!(\alpha'-\beta'-1)!}{\gamma'!(\gamma'-1)!(\alpha'-\beta'-\gamma')!}}_{=L(\alpha'-\beta',\gamma')}(2\beta')^{\overline{\gamma'}}\\
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&=\frac{2^{2\beta'-\alpha'}(-1)^{\alpha'-\beta'}}{(\alpha'-\beta')!(\alpha'-\beta'-1)!}\sum_{\gamma'=\min\{\alpha'-2\beta',\,0\}}^{\alpha'-\beta'}\frac{2^{\gamma'}(\gamma'-1)!}{(2\beta'-\alpha'+\gamma')!}(-1)^{\gamma'}L(\alpha'-\beta',\gamma')(2\beta')^{\overline{\gamma'}}
\end{align*}
The term $\frac{2^{\gamma'}(\gamma'-1)!}{(2\beta'-\alpha'+\gamma')!}$ and the lower bound $\min\{\alpha'-2\beta',\,0\}$ for the sum are annoying ; here we have something pretty close to
$$\sum_{\gamma'=0}^{\alpha'-\beta'}(-1)^{\gamma'}L(\alpha'-\beta',\gamma')(2\beta')^{\overline{\gamma'}}=(2\beta')^{\underline{\alpha'-\beta'}}.$$
Here $x^{\underline{k}}:=x(x-1)\ldots(x-k+1)$ denotes the falling factorial.
2.2. Second attempt. We could as well write:
\begin{align*}
S(\alpha',\beta')&=\sum_{\gamma=0}^{\min\{\beta',\,\alpha'-\beta'\}}\frac{2^{\beta'-\gamma}(-1)^\gamma}{\gamma!(\beta'-\gamma)!}\frac{2\beta'(2\beta'+1)\ldots(\alpha'+\beta'-\gamma)}{(\alpha'-\beta'-\gamma)!}\\
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&=\frac{2}{(\beta'-1)!}\sum_{\gamma=0}^{\min\{\beta',\,\alpha'-\beta'\}}2^{\beta'-\gamma}(-1)^\gamma\binom{\beta'}{\gamma}\binom{\alpha'+\beta'-\gamma}{\alpha'-\beta'-\gamma}.
\end{align*}
Now this looks like a Vandermonde's identity: https://en.wikipedia.org/wiki/Vandermonde%27s_identity. This time the annoying term is $2^{\beta'-\gamma}(-1)^\gamma$.
EDIT. After having read some comments/answers, I think I can handle my problem IF I can get a bound such as $B_{n,k}(2!,3!,\ldots,(n-k+2)!)\leq n!$ or $-$ even better $-$ $B_{n,k}(2!,3!,\ldots,(n-k+2)!)\leq(n-k+1)!$, where I computed:

\begin{align*}
B_{n,k}(2!,\ldots,(n-k+2)!)&=\sum_{j=0}^{\min\{k,\,n-k\}}\binom{k}{j}2^{k-j}(-1)^j\frac{(2k)^{(n-k-j)}n!}{(n-k-j)!k!}.
\end{align*}

 A: For $n\ge k\ge0$, the Bell polynomials of the second kind $B_{n,k}$ satisfy
\begin{equation}\label{Bell-2!-3!}\tag{+}
B_{n,k}(2!,3!,\dotsc,(n-k+2)!)
=\frac{n!}{k!}\sum_{\ell=0}^k(-1)^{k-\ell} \binom{k}{\ell}\binom{n+2\ell-1}{n},
\end{equation}
where $\binom{q}{0}=1$ for all $q\in\mathbb{C}$.
The formula \eqref{Bell-2!-3!} was cited in Lemma 3.4 of the paper [1], reviewed in Section 1.8 (pp. 8--9) of the paper [2], and proved in Lemma 6 of the paper [3] below.
References

*

*F. Qi and B.-N. Guo, Explicit and recursive formulas, integral representations, and properties of the large Schroder numbers, Kragujevac J. Math. 41 (2017), no. 1, 121--141; available online at https://doi.org/10.5937/KgJMath1701121F.

*Feng Qi, Da-Wei Niu, Dongkyu Lim, and Yong-Hong Yao, Special values of the Bell polynomials of the second kind for some sequences and functions, Journal of Mathematical Analysis and Applications 491 (2020), no. 2, Paper No. 124382, 31 pages; available online at https://doi.org/10.1016/j.jmaa.2020.124382.

*F. Qi, X.-T. Shi, and B.-N. Guo, Two explicit formulas of the Schroder numbers, Integers 16 (2016), Paper No. A23, 15 pages.

