This is so fascinating ... inspired by Gottfried's investigations I was playing around with the logit of other functions, namely \(x\mapsto xe^x\), \(x\mapsto x+\frac{1}{2}x^2\), \(x\mapsto x+\frac{1}{3}x^2\) and \(x\mapsto x+3x^2+7x^3\). It looks like this sinoidal pattern only depends on the coefficient \(c\) of \(x^2\) !
It seems that the growth has the form
\[ a_k = \frac{(k-3)!}{\left(\frac{2\pi}{c}\right)^k} \]
i.e. \(j_k/a_k\) shows this sinoidal pattern. I did not yet investigate the cases \(c=0\) ...
Here is the numerical evidence:
It seems that the growth has the form
\[ a_k = \frac{(k-3)!}{\left(\frac{2\pi}{c}\right)^k} \]
i.e. \(j_k/a_k\) shows this sinoidal pattern. I did not yet investigate the cases \(c=0\) ...
Here is the numerical evidence:
