No, $F$ is not onto in general.  For example, it is not difficult to show that, when $n=3$, the polynomial
$$
c = c_0 + c_1t + c_2 t + c_3 t^3 + c_4 t^4
$$
is in the image of $F$ if and only if 
$$
12\,c_0c_4 - 3\,c_1c_3 + {c_2}^2 \ge 0.
$$