sequences and series - Is the “almost-identity” $sum_{k=0}^infty left[pi^{frac k2}big/Gamma{left(frac k2+1right)}right]approx46$ significant or a coincidence?

Recently, I read an article about “almost-identities”. It said, that for every “almost-identity” we have to decide whether it is a coincidence or not. By myself, I discovered that
$$\sum_{k=0}^\infty \frac{\pi^{\frac k2}}{\Gamma{\left(\frac k2+1\right)}}=e^{\pi}\left(1+\text{erf}\left(\sqrt\pi\right)\right)\approx45.9993260894...$$
which is surprisingly close to $46$. So my question is: is this a mere coincidence or can it be “proven” in some sense?