Powers of Jacobi triple product, Cohen's numbers and the Ramanujan $\mathrm{\Delta }$-function

We show that the eighth power of the Jacobi triple product is a Jacobi–Eisenstein series of weight $4$ and index $4$ and we calculate its Fourier coefficients. As applications we obtain explicit formulas for the eighth powers of theta-constants of arbitrary order and the Fourier coefficients of the Ramanujan Delta-function $\mathrm{\Delta }(\tau )={\eta }^{24}(\tau )$, ${\eta }^{12}(\tau )$ and ${\eta }^8(\tau )$ in terms of Cohen’s numbers $H(3,N)$ and $H(5,N)$. We give new formulas for the number of representations of integers as sums of eight higher figurate numbers. We also calculate the sixteenth and the twenty-fourth powers of the Jacobi theta-series using the basic Jacobi forms.