Series and integral representations of the Taylor coefficients of the Weierstrass sigma-function

We provide two kinds of representations for the Taylor coefficients of the Weierstrass $\sigma$-function $\sigma (\cdot ;\mathrm{\Gamma })$ associated to an arbitrary lattice $\mathrm{\Gamma }$ in the complex plane $\mathbb{C}={\mathbb{R}}^2$ - the first one in terms of the so-called Hermite-Gauss series over $\mathrm{\Gamma }$ and the second one in terms of Hermite-Gauss integrals over $\mathbb{C}$.