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

Recursions for rational q,t-Catalan numbers