Let $ c_{n,k} $ be the Simsun permutations$^1$ defined by the following relations: $\displaystyle c_{n,0} = 1, \hspace{0.1cm} (n \ge 1);$ $$ c_{n,k} = (k+1) c_{n-1,k} +(n-2k+1) c_{n-1,k-1}, \hspace{0.5cm} (1 \leq k \leq \lfloor n/2 \rfloor);$$ and $ c_{n,k} = 0, \hspace{0.1cm} ( k> \lfloor n/2 \rfloor). $
Now, let $n=2p.$ I am trying to find the value of \begin{eqnarray} A_p:=\sum_{k=0}^{p} c_{2p,k} \hspace{0.1cm} \frac{(2p-2k)!}{2^{p-k} (p-k)!}. \end{eqnarray} or at least a sharp upper bound for it. We know$^2$ that $$ \sum_{k=0}^{p} c_{2p,k} \hspace{0.1cm} 2^{2p-k} = (2p+1)!.$$
Motivation: In a part of my research (in quantum statistical mechanics) I need to show convergence of a series. I have reduced the initial problem to finding the value of $ A_p $, or at least a good upper bound for $ A_p $.
Any hint or idea would be appreciated! Thanks in advance!