In my attempt to complete this answer, I hit a snag in showing that
$\lim_{t\to 0} \dfrac{\mathrm d^{k-1}}{\mathrm dt^{k-1}}\left(\frac{t\sqrt{1+t}}{\log\sqrt{1+t}}\right)^k=2(k+2)^{k-1}$
This shows up when trying to apply Lagrangian inversion to the function $\dfrac{\log\sqrt{1+x}}{\sqrt{1+x}}$. My sticking point here is that I am unable to find a convenient expression for the derivatives. Is there an easy proof for this?