Consider a smooth plane curve $X\subset\mathbb{P}^2$ of degree $d$. We will say that $x\in X$ is an inflection point of order $s$ if the tangent line $T_xX$, of $X$ at $x\in X$, intersects $X$ in $x\in X$ with multiplicity at least $s$.

For instance, any point of $X$ in an inflection point of order $s = 2$, and the inflection points of order $s = 3$ are the flexes of $X$.

Does there exist a closed formula for the number $I(s)$ of inflection points of order $s$ of $X$. For instance, it is well known that $I(3) = 3d(d-2)$ since flexes are the intersection points of $C$ with its Hessian which is a curve of degree $3(d-2)$.

Furthemore, we must have $I(s) = 0$ for $s > d$ since otherwise $C$ would be reducible.