I've learned that $$\ln\left(1+\cfrac{x}{y}\right)=\frac{x}{y+\cfrac{1x}{2+\cfrac{1x}{3y+\cfrac{2x}{2+\cfrac{2x}{5y+\frac{3x}{2+\cdots}}}}}}=\cfrac{2x}{2y+x-\cfrac{(1x)^2}{3(2y+x)-\cfrac{(2x)^2}{5(2y+x)-\cfrac{(3x)^2}{7(2y+x)-\cdots}}}}$$[reference]
I noticed that the pattern of the coefficients in the numerators, $\{1,1,2,2,3,3,\dots\}$, matches a continued fraction that I'm studying. It has the form $$y+\cfrac{1}{z+\cfrac{1}{y+\cfrac{2}{z+\cfrac{2}{y+\cfrac{3}{z+\cdots}}}}}$$ I'm not interested in a closed form. Although it would be interesting to see what you come up with. But my main question is, can we translate this into a form like the one in the RHS above where the numerators follow a more natural pattern? $\{1,2,3,4,5,\dots\}$ instead of $\{1,1,2,2,3,3,4,4,\dots\}$