sequences and series - Looking for a closed form for $a_m =sum_{k=0}^{infty}binom{k+m}{m}frac{1}{4^{k}(2(k+m))!}$

I have this sequence
$$
a_m =\sum_{k=0}^{\infty}\binom{k+m}{m}\frac{1}{4^{k}(2(k+m))!}
$$
and there seems to exist a patern arising when it is evaluated by WA. It involves $\cosh(1/2)$ and $\sinh(1/2)$. Bellow are the first
$$

\begin{align*}
a_0 &= \cosh\left(\frac{1}{2}\right)\\
a_1 &= \sinh\left(\frac{1}{2}\right)\\
a_2 &= \frac{1}{2}\left(\cosh\left(\frac{1}{2}\right )-2\sinh\left(\frac{1}{2} \right ) \right )\\
&\vdots\\
\end{align*}
$$
but $\cosh(1/2)$ and $\sinh(1/2)$ keeps showing up for $a_3$, $a_4$, $\cdots$

Could anyone find a general expression for $a_m$ involving $\cosh(1/2)$ and $\sinh(1/2)$.

I'd apprecaite any help, thanks.

Answer

$$a_m = \sum_{k\geq 0}\frac{(k+m)! 4^{-k}}{m! k! (2k+2m)!}=\frac{4^m}{m!}\cdot\left.\frac{d^m}{dx^m}\sum_{k\geq 0}\frac{(x/4)^{k+m}}{(2k+2m)!}\right|_{x=1} $$
and now you may recognize in the last series part of the Taylor series of $\cosh(\sqrt{x}/2)$.