Mathematica seems to offer closed forms for some of the expressions above. For example \[\underset{k=1}{\overset{\infty}{\large K \normalsize}} \frac{1}{x} = \frac{2}{x+\sqrt{4+x^2}}\\ \underset{k=1}{\overset{\infty}{\large K \normalsize}} \frac{n}{x} = \frac{2n}{x+\sqrt{4n+x^2}}\] where we then see that the Laurent expansion does indeed follow the form \[\lim_{x\to\infty}\frac{2}{x+\sqrt{4+x^2}}=\sum_{k=1}^\infty \frac{C(k)}{x^{2k+1}}\] with \(C(k)\) the Catalan numbers.