  We give a simple recursion labeled by binary sequences which computes
rational $q,t$-Catalan power series, both in relatively prime and non
relatively prime cases. It is inspired by, but not identical to recursions due
to B. Elias, M. Hogancamp, and A. Mellit, obtained in their study of link
homology. We also compare our recursion with the Hogancamp-Mellit's recursion
and verify a connection between the Khovanov-Rozansky homology of $N,M$-torus
links and the rational $q,t$-Catalan power series for general positive $N,M.$
