Show that all Eisenstein series $G_k$ can be expressed as polynomials in $G_4$ and $G_6$, e.g. express $G_8$ and $G_{10}$ in this way. Hint: Setting $a_n := (2n+1)G_{2n+2}$, show that $2n(2n-1)a_n = 6(2a_n + \sum\limits_{k=1}^{n-2} a_{k}a_{n-1-k})$ , $n>2$
Obviously, what I have trouble proving, is the Hint. It seems, like I should use the second derivative of Weierstrass function representation, using the Eisenstein series (namely $\wp = \displaystyle\frac{1}{z^2} + \sum\limits_{m=1}^{\infty} (2m+1)G_{2m+2}z^{2m}$), but I'm not sure where the sum on the right comes from.
Any help will be appreciated.
EDIT: Can someone help some more?