Some defective secant varieties to osculating varieties of Veronese surfaces

We consider the $k$-osculating varieties $O_{k,d}$ to the Veronese $d-$uple embeddings of $mathbb{P}^{2}$. By studying the Hilbert function of certain zero--dimensional schemes $Ysubset mathbb{P}^2$, we find the dimension of $O^s_{k,d}$, the $(s-1)^{th}$secant varieties of $O_{k,d}$, for $3 leq sleq 6$ and $s=9$, and we determine whether those secant varieties are defective or not.