Extensions of an identity of Chan and Cooper in the spirit of Ramanujan

Chan and Cooper proved that if the integers \({c(n) (n=0,1,2,\ldots )}\) $c(n)(n=0,1,2,\dots )$ are given by \(\begin{aligned} \sum _{n=0}^\infty c(n)q^n = \prod _{n=1}^\infty \frac{1}{\left( 1-q^n\right) ^2\left( 1-q^{3n}\right) ^2}, \end{aligned}\) $$\begin{array}{c}\hfill \sum _{n=0}^{\infty }c(n)q^n=\prod _{n=1}^{\infty }\frac{1}{{\left(1,-,q^n\right)}^2{\left(1,-,q^{3n}\right)}^2},\end{array}$$ then \(\begin{aligned} \sum _{n=0}^\infty c(2n+1)q^n = 2 \prod _{n=1}^\infty \frac{\left( 1-q^{2n}\right) ^4\left( 1-q^{6n}\right) ^4}{\left( 1-q^n\right) ^6\left( 1-q^{3n}\right) ^6}. \end{aligned}\) $$\begin{array}{c}\hfill \sum _{n=0}^{\infty }c(2n+1)q^n=2\prod _{n=1}^{\infty }\frac{{\left(1,-,q^{2n}\right)}^4{\left(1,-,q^{6n}\right)}^4}{{\left(1,-,q^n\right)}^6{\left(1,-,q^{3n}\right)}^6}.\end{array}$$ We prove many other results of this type and apply them to the determination of congruence properties of the coefficients.