Recursive prescription for logarithmic jet rate coefficients

We derive a recursion relation for the analytic leading logarithmic coefficients of a final state gluon cascade. We demonstrate the potential of our method by analytically computing the rate coefficients for the emission of up to 80 gluons in both the exclusive-${k}_{t}$ (Durham) and generalized inclusive-${k}_{t}$ class of jet algorithms. There is a particularly simple form for the ratios of resolved coefficients. We suggest potential applications for our method including the efficient generation of shower histories.