Consider the degree two Veronese embedding $\mathbb{P}^n\rightarrow\mathbb{P}^N$ and let $V^n_{2}\subset\mathbb{P}^N$ be the corresponding Veronese variety.

Let $Sec_k(V^n_{2})\subseteq\mathbb{P}^N$ be the $k$-secant variety of $V_{2}^{n}$. This is the closure of the union of all $(k-1)$-planes spanned by $k$ independent points on $V_2^n$.

Is there a closed formula for the degree of $Sec_k(V^n_{2})$?

For instance if $k = 1$ we have that $Sec_1(V_2^n) = V_2^n$ has degree $2^n$, while for $k = n$ we have that $Sec_n(V_2^n)\subset\mathbb{P}^N$ is a hypersurface of degree $n+1$. What about the degree of $Sec_k(V^n_{2})\subseteq\mathbb{P}^N$ for $1 < k < n$?