$$a_k={\left(-1\right)}^k\sqrt{\frac{2}{\pi }}{e}^{g}k\sum _{j=0}^k{\left(-1\right)}^j\frac{\left(k+j-1\right)!}{\left(k-j\right)!j!}{\left(\frac{e}{j+g+\frac{1}{2}}\right)}^{j+\frac{1}{2}}$$