Laurent coefficients of the Weierstrass elliptic function

Writing $G_{2r}={\sum }_{\omega \ne 0}{\omega }^{-2r}$ for the lattice Eisenstein sums, expansion about zero givesThus the constant coefficient vanishes and, for $k\ge 1$, the coefficient of $z^{2k}$ is $(2k+1)G_{2k+2}$.

Write the Weierstrass elliptic function as $\wp (z)=z^{-2}+{\sum }_{n\ge 1}{c}_n{z}^{2n}$, 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 ${\wp }^{\prime \prime }=6{\wp }^2-g_2/2$. Comparing coefficients of $z^{2n-2}$ gives the displayed recurrence for $n\ge 3$. 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 $G_{10}=5G_4{G}_6/11$.