Milnor invariants of length 2k+2 for links with vanishing Milnor invariants of length ≤k

J.-B. Meilhan and the second author showed that any Milnor μ¯-invariant of length between 3 and 2k+1 can be represented as a combination of HOMFLYPT polynomial of knots obtained by certain band sum of the link components, if all μ¯-invariants of length ≤k vanish. They also showed that their formula does not hold for length 2k+2. In this paper, we improve their formula to give the μ¯-invariants of length 2k+2 by adding correction terms. The correction terms can be given by a combination of HOMFLYPT polynomial of knots determined by μ¯-invariants of length k+1. In particular, for any 4-component link the μ¯-invariants of length 4 are given by our formula, since all μ¯-invariants of length 1 vanish.