= Positive polynomial recurrence for lattice Eisenstein sums
{title2=$c_n=\frac{3}{(2n+3)(n-2)}\sum_{r=1}^{n-2}c_rc_{n-1-r}$}
Write the <Weierstrass elliptic function> as $\wp(z)=z^{-2}+\sum_{n\geq1}c_nz^{2n}$, with $c_1=3G_4$, $c_2=5G_6$ and $c_n=(2n+1)G_{2n+2}$. Differentiate the <Weierstrass elliptic differential equation> to obtain $\wp''=6\wp^2-g_2/2$. Comparing coefficients of $z^{2n-2}$ gives the displayed recurrence for $n\geq3$. Every denominator is positive, so induction expresses every higher <lattice Eisenstein series> as a polynomial in $G_4,G_6$ with positive rational coefficients on its occurring monomials. In particular $G_8=3G_4^2/7$ and $G_{10}=5G_4G_6/11$.
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