Jacobi cubic product identity (${\prod }_{n\ge 1}(1-q^n{)}^3={\sum }_{r\ge 0}(-1{)}^r(2r+1)q^{r(r+1)/2}$)

The odd Dirichlet character modulo four has ${\chi }_4(2r+1)=(-1{)}^r$. The series $B=\frac{1}{2}{\sum }_{n}n{\chi }_4(n)q^{n^2/8}$ is therefore $q^{1/8}$ times the displayed sum. The character theta proof of the eta product gives $B={\eta }^3$, proving the identity.