<s>[INST] How can I simplify this sum any further? Recently I was playing around with some numbers and I stumbled across the following formal power series:
$$\sum_{k=0}^\infty\frac{x^{ak}}{(ak)!}\biggl(\sum_{l=0}^k\binom{ak}{al}\biggr)$$
I was able to "simplify" the above expression for $a=1$:
$$\sum_{k=0}^\infty\frac{x^k}{k!}\cdot2^k=e^{2x}$$
I also managed to simplify the expression for $a=2$ with the identity $\sum_{i=0}^\infty\frac{x^{2k}}{(2k)!}=\cosh(x)$:
$$\sum_{k=0}^\infty\frac{x^{2k}}{(2k)!}\biggl(\sum_{l=0}^k\binom{2k}{2l}\biggr)=\mathbf[\cdots\mathbf]=\frac{1}{4}\cdot(e^{2x}+e^{-2x})+\frac{1}{2}=\frac{1}{2}\cdot(\cosh(2x)+1)$$
However, I couldn't come up with a general method for all $a\in\Bbb{N}$. I would be very thankful if someone could either guide me towards simplifying this expression or post his solution here. 
 [/INST] You might be able to use the fact that
$$\sum_{k=0}^\infty b_{ak}=\sum_{k=0}^\infty \left(\frac{1}{a}\sum_{j=0}^{a-1} \exp\left(2\pi ijk/a\right)\right)b_k.$$
For example, when $a=1$, taking $b_k = \frac{x^k}{k!}\sum_{\ell \ge 0} \binom{k}{\ell}$ yields
$$\sum_{k=0}^\infty b_{k}=\sum_{k=0}^\infty \frac{x^k}{k!}\sum_{\ell \ge 0} \binom{k}{\ell}=\sum_{k=0}^\infty \frac{x^k}{k!}2^k=\exp(2x),$$
as you already obtained.
For $a=2$, first note that
\begin{align}
\sum_{\ell \ge 0}\binom{k}{2 \ell} &= \sum_{\ell\ge 0} \left(\frac{1}{2}\sum_{j=0}^1 \exp\left(2\pi ij\ell/2\right)\right)\binom{k}{\ell}\\
&= \sum_{\ell\ge 0} \frac{1+(-1)^\ell}{2}\binom{k}{\ell}\\
&= \frac{1}{2}\sum_{\ell\ge 0} \binom{k}{\ell}+ \frac{1}{2}\sum_{\ell\ge 0} (-1)^\ell\binom{k}{\ell}\\
&= \frac{2^k+0^k}{2}.
\end{align}
Now taking $b_k = \frac{x^k}{k!}\sum_{\ell \ge 0}\binom{k}{2 \ell}$ yields
\begin{align}\sum_{k=0}^\infty b_{2k}&=\sum_{k=0}^\infty \left( \frac{1}{2}\sum_{j=0}^1 \exp\left(\pi ijk\right)\right)\frac{x^k}{k!}\sum_{\ell \ge 0}\binom{k}{2 \ell}\\
&=\frac{1}{2}\sum_{k=0}^\infty \left(1+(-1)^k\right)\frac{x^k}{k!}\left(2^{k-1}+\frac{1}{2}[k=0]\right)\\
&=\frac{1}{4}\sum_{k=0}^\infty \frac{x^k}{k!}2^k+\frac{1}{4}\sum_{k=0}^\infty (-1)^k\frac{x^k}{k!}2^k+\frac{1}{2}\\
&=\frac{\exp(2x)+\exp(-2x)}{4} +\frac{1}{2}\\
&=\cosh^2(x),
\end{align}
again matching your result.
For $a=3$, first note that
\begin{align}
\sum_{\ell \ge 0}\binom{k}{3 \ell} &= \sum_{\ell\ge 0} \left(\frac{1}{3}\sum_{j=0}^2 \exp\left(2\pi ij\ell/3\right)\right)\binom{k}{\ell}\\
&= \sum_{\ell\ge 0} \frac{1+\exp(2\pi i\ell/3)+\exp(4\pi i\ell/3)}{3}\binom{k}{\ell}\\
&= \frac{1}{3}\sum_{\ell\ge 0} \binom{k}{\ell}+ \frac{1}{3}\sum_{\ell\ge 0} \exp(2\pi i/3)^\ell\binom{k}{\ell}+ \frac{1}{3}\sum_{\ell\ge 0} \exp(4\pi i/3)^\ell\binom{k}{\ell}\\
&= \frac{2^k+(1+\exp(2\pi i/3))^k+(1+\exp(4\pi i/3))^k}{3}\\
&= \frac{2^k+\exp(\pi i/3)^k+\exp(-\pi i/3)^k}{3}.
\end{align}
Now taking $b_k = \frac{x^k}{k!}\sum_{\ell \ge 0}\binom{k}{3 \ell}$ yields
\begin{align}\sum_{k=0}^\infty b_{3k}&=\sum_{k=0}^\infty \left( \frac{1}{3}\sum_{j=0}^2 \exp\left(2\pi ijk/3\right)\right)\frac{x^k}{k!}\sum_{\ell \ge 0}\binom{k}{3 \ell}\\
&=\frac{1}{3}\sum_{k=0}^\infty \left(1+\exp(2\pi ik/3)+\exp(4\pi ik/3)\right)\frac{x^k}{k!}\frac{2^k+\exp(\pi i/3)^k+\exp(-\pi i/3)^k}{3}\\
&=\frac{1}{9}\sum_{k=0}^\infty (1+\exp(2\pi i/3)^k+\exp(4\pi i/3)^k)(2^k+\exp(\pi i/3)^k+\exp(-\pi i/3)^k)\frac{x^k}{k!}.
\end{align}
Now expand the product of trinomials to obtain 9 sums that reduce to $\exp(cx)$ for various constants $c$.
Alternatively, note that:
$$\sum_{k=0}^\infty \frac{x^{ak}}{(ak)!}\sum_{\ell \ge 0}\binom{ak}{a\ell} = \left(\sum_{k=0}^\infty \frac{x^{ak}}{(ak)!}\right)^2,$$
so you might as well just compute
\begin{align}
\sum_{k=0}^\infty \frac{x^{ak}}{(ak)!}
&= \sum_{k=0}^\infty \left( \frac{1}{a}\sum_{j=0}^{a-1} \exp\left(2\pi ijk/a\right)\right)\frac{x^k}{k!} \\
&= \frac{1}{a}\sum_{j=0}^{a-1} \sum_{k=0}^\infty \frac{(\exp(2\pi ij/a)x)^k}{k!} \\
&= \frac{1}{a}\sum_{j=0}^{a-1} \exp(\exp(2\pi ij/a)x),
\end{align}
and then square the result.
</s>