On Series Involving Cubed Catalan Numbers

Using generalized binomial coefficient identities and some results of John Dougall, we derive some families of series involving the cubes of Catalan numbers. We also establish a family of series containing fourth powers of Catalan numbers. Finally, we find a generalization of the Bauer series for $1/\pi$ and obtain some Ramanujan-like series for $1/\pi^2$ and~$1/\pi^3$.