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 $$ \sum_{k=0}^{p} c_{2p,k} \hspace{0.1cm} 2^{2p-k} = (2p+1)!.$$

We also know$^2$ that $ \sum_{k=0}^{p} c_{2p,k} = T_{p+1}$, where $  T_n= \frac{2^{2n}(2^{2n}-1) |B_{2n}|}{2n} $ is the sequence of tangent numbers$^3$ (defined by the Bernoulli numbers $B_n$), appearing in the Taylor series expansion of tan($x$): $$\text{tan}(x)=\sum_{n=1}^{\infty} T_n \frac{x^{2n-1}}{(2n-1)!}.$$

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 greatly appreciated! Thanks in advance!


 
1. For Andre and Simsun permutations see [here][1] and [here][2].
2. See [here][3] for the paper "increasing trees and alternating permutations" by G. Kuztensov, I. Pak, and A.E. Postnikov. In this paper, the Andre permutations are denoted by $ d_{n,k} $.
3. See [here][4].


  [1]: http://oeis.org/A094503
  [2]: http://oeis.org/A113897
  [3]: https://www.google.ca/webhp?sourceid=chrome-instant&ion=1&espv=2&ie=UTF-8#q=increasing%20trees%20and%20alternating%20permutations%20Kuznetsov.
  [4]: http://oeis.org/A000182