General polynomial formula for the coefficient polynomials

Derive a general explicit expression for the integer polynomials whose zeros are the coefficients c_{k,i} appearing in the linear-combination formula for the Fibonacci-like sequence A_{k,n}.

Background

The paper expresses the Fibonacci-like sequence A_{k,n} as a linear combination of powers of the zeros r_{k,i} of Q_k(X)=X^k-X^{k-1}-1, with coefficients c_{k,i}. It then observes that, for each fixed k, these coefficients are themselves the zeros of an integer polynomial of degree k. The appendix proves the existence and determines the leading and constant coefficients of these polynomials, while several initial examples are listed for small values of k.

What remains unresolved is an explicit general formula that directly constructs these coefficient polynomials for arbitrary k, rather than merely establishing their existence through symmetric-polynomial arguments. Such a formula would clarify the algebraic structure of the coefficients c_{k,i} and extend the small-k examples presented in the paper.