A recursive formula for osculating curves

Let $X$ be a smooth complex projective variety. Using construction devised to Gathmann, we present recursive formula for some of the Gromov-Witten invariants $X$. We prove that, when is homogeneous, this gives number osculating rational curves at general point hypersurface This generalizes classical well known pairs inflexion (asymptotic) lines surfaces in $\mathbb{P}^{3}$ Salmon, as Darboux's $27$ conics.