<s>[INST] Power Series involving Double Factorials This comes from my partial solution to another question. I need to find a closed form for the following summation

$$\sum_{n=1}^{\infty}\frac{n!}{(2n+1)!!}n^x$$
  Where $x$ is a fixed integer

This appears similar to many arcsine series and formulas

Particular values:
$$\begin{array}{c|c|c|} 
 & 0 & 1 & 2 & 3 & 4 \\ \hline
x & \frac{\pi}{2}-1 & \frac{0\pi}{2}+1 & \frac{\pi}{2}+1 & \frac{3\pi}{2}+5 & \frac{16\pi}{2}+25\\ \hline
\end{array}$$
$$\begin{array}{c|c|c|} 
 & 5 & 6 & 7 & 8 \\ \hline
x & \frac{105\pi}{2}+165 & \frac{841\pi}{2}+1321 & \frac{7938\pi}{2}+12469 & \frac{86311 π}{2}+135577\\ \hline
\end{array}$$

Update 1: @RobertIsrael found a closed form for this sum's Exponential Generating Function, but I have been unable to find any expression for the Maclaurin series.
Update 2: Grant B. has found a simple recurrence, but a complete closed form has not been found
 [/INST] Why just even powers of $n$?  Let
$$ F(j) = \sum_{n=1}^\infty \frac{n! n^j}{(2n+1)!!} = \sum_{n=1}^\infty \frac{2^{-1-n} \sqrt{\pi} n! n^j}{\Gamma(n+3/2)} $$ 
The exponential generating function is
$$\eqalign{E(z) &= \sum_{j=0}^\infty \frac{F(j)}{j!} z^j\cr
                &= \sum_{n=1}^\infty \frac{2^{-1-n} \sqrt{\pi} n! e^{zn}}{\Gamma(n+3/2)}\cr
&= -1 + \frac{2\; e^{-z/2}}{\sqrt{2-e^z}} \arcsin\left(e^{z/2}/\sqrt{2}\right)}$$
EDIT: The last line could be obtained (in hindsight) as follows.  As you know, (for $|x| < 1$)
$$\arcsin(x) = \sum _{k=0}^{\infty }{\frac { \left( 2\,k \right) !\,{4}^{-k}{x}^{2\,k
+1}}{ \left( k! \right) ^{2} \left( 2\,k+1 \right) }}
$$
Taking $x = e^{z/2}/\sqrt{2}$, we want to multiply this by 
$$ \frac{2 e^{-z/2}}{\sqrt{2-e^z}} = \frac{1}{x \sqrt{1-x^2}} =  \sum_{k=0}^\infty \frac{(2k)!\, 4^{-k} x^{2k-1}}{(k!)^2}$$
The Cauchy product is 
$$ \sum_{n=0}^\infty \sum_{k=0}^n \frac{(2k)! 4^{-k}}{(k!)^2} \frac{(2(n-k))! 4^{-n+k}}{((n-k))!^2(2n-2k+1)} x^{2n}$$ 
And it turns out that
$$ \sum _{k=0}^{n}{\frac { \left( 2\,k \right) !\, \left( 2\,n-2\,k
 \right) !}{ \left( k! \right) ^{2} \left(  \left( n-k \right) !
 \right) ^{2}(2n-2k+1)}}={\frac {\sqrt {\pi}{4}^{n}n!}{2\,\Gamma
 \left( n+3/2 \right) }}
$$
</s>