\[\sum_{n=0}^{\infty}\frac{q^{n^{2}+2n}}{\left(q^{2};q^{2}\right)_{n}\left(q;q^{%
2}\right)_{n+1}}=\mbox{ coeff. of }z^{0}\mbox{ in }\frac{\left(-zq;q^{2}\right%
)_{\infty}\left(-z^{-1}q;q^{2}\right)_{\infty}\left(q^{2};q^{2}\right)_{\infty%
}}{\left(-q^{2}z^{-1};q^{2}\right)_{\infty}\left(q;q^{2}\right)_{\infty}\left(%
z^{-1}q^{2};q^{2}\right)_{\infty}}=\frac{1}{\left(q;q^{2}\right)_{\infty}}%
\mbox{ coeff. of }z^{0}\mbox{ in }\frac{\left(-zq;q^{2}\right)_{\infty}\left(-%
z^{-1}q;q^{2}\right)_{\infty}\left(q^{2};q^{2}\right)_{\infty}}{\left(q^{4}z^{%
-2};q^{4}\right)_{\infty}}=\frac{H(q^{4})}{\left(q;q^{2}\right)_{\infty}}.\]