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G. Mikhalkin introduced a refined count for real rational curves in a toricsurface which pass through some points on the toric boundary of the surface.The refinement is provided by the value of a so-called quantum index. Moreover,he proved that the result of this refined count does not depend on the choiceof the points. The correspondence theorem allows one to compute theseinvariants using the tropical geometry approach and the refinedBlock-G\"ottsche multiplicities. In this paper we give a recursive formula forthese invariants, that leads to an algorithm to compute them.

A Caporaso-Harris type Formula for relative refined invariants