${a}_{n,k}|n\to$ Math_356# ${K}_{\frac{1}{2}-k}$ Expression for ${K}_{\frac{1}{2}-k}$ k = 0 ${K}_{\frac{1}{2}}$ 1 $\sqrt{\pi }{\text{e}}^{\frac{-x}{2}}{x}^{-\frac{1}{2}}$ k = 1 ${K}_{-\frac{1}{2}}$ 1 2 $\sqrt{\pi }{\text{e}}^{\frac{-x}{2}}{x}^{-\frac{1}{2}}\left(\frac{2}{x}+1\right)$ k = 2 ${K}_{-1\frac{1}{2}}$ 1 6 12 $\sqrt{\pi }{\text{e}}^{\frac{-x}{2}}{x}^{-\frac{1}{2}}\left(\frac{12}{{x}^{2}}+\frac{6}{x}+1\right)$ k = 3 ${K}_{-2\frac{1}{2}}$ 1 12 60 120 $\sqrt{\pi }{\text{e}}^{\frac{-x}{2}}{x}^{-\frac{1}{2}}\left(\frac{120}{{x}^{3}}+\frac{60}{{x}^{2}}+\frac{12}{x}+1\right)$ k = 4 ${K}_{-3\frac{1}{2}}$ 1 20 180 840 1680 $\sqrt{\pi }{\text{e}}^{\frac{-x}{2}}{x}^{-\frac{1}{2}}\left(\frac{1680}{{x}^{4}}+\frac{840}{{x}^{3}}+\frac{180}{{x}^{2}}+\frac{12}{x}+1\right)$ k = 5 ${K}_{-4\frac{1}{2}}$ 1 30 420 3360 4120 30,240 $\sqrt{\pi }{\text{e}}^{\frac{-x}{2}}{x}^{-\frac{1}{2}}\left(\frac{1680}{{x}^{4}}+\frac{840}{{x}^{3}}+\frac{180}{{x}^{2}}+\frac{12}{x}+1\right)$ ${x}^{0}$ ${x}^{-1}$ ${x}^{-2}$ ${x}^{-3}$ ${x}^{-4}$ ${x}^{-5}$ $\sqrt{\pi }{\text{e}}^{\frac{-x}{2}}{x}^{-\frac{1}{2}}\left(\frac{30240}{{x}^{4}}+\frac{4120}{{x}^{4}}+\frac{33160}{{x}^{3}}+\frac{420}{{x}^{2}}+\frac{30}{x}+1\right)$