Closed form for $\left(-1\right)^{n}\sum_{k=0}^{n}\left(-1\right)^{k}\binom{n}{k}2^{\binom{k}{2}}$
There is no closed form for $a_n$.
However, it seems that $$\log(a_n) \sim C x^2 \qquad \text{with} \qquad C \sim 0.34614$$ This results from a quick and dirty regression for $3 \leq n \leq 1000$ $(a_{1000} \sim 3.04 \times 10^{150364})$.
The constant $C$ was not recognized by inverse symbolic calculators but it looks like $\frac 12 \log(2)$.
Notice that, in the $OEIS$ page, Vaclav Kotesovec proposed $$a_n \sim 2^{\frac{n(n-1)}{2} }$$ which is in a relative arror of $1$% if $n>11$, $0.1$% if $n>15$, $0.01$% if $n >18$