Combinatorial significance of the remaining invariants

Determine the combinatorial significance and properties of the invariants a_{j,k} associated with the recursive polynomial sequence p_0(x)=x^2-2 and p_n(x)=p_{n-1}(x)^2-2, excluding the diagonal invariants a_{k,k} whose weighted Catalan-number representation is established.

Background

The paper studies the coefficients of the recursively defined polynomials p_n(x)=p_{n-1}(x)^2-2 and expresses each coefficient c_{n,2k} as a linear combination of powers 2^{2jn} with coefficients a_{j,k} that are independent of n. These constants are invariants of the polynomial sequence.

The paper identifies the diagonal invariants a_{k,k} as weighted Catalan numbers through a representation involving labeled ordered trees. However, it does not provide a combinatorial interpretation for the other invariants a_{j,k}, leaving their significance as an explicit unresolved question.