Convoluted Convolved Fibonacci Numbers

The convolved Fibonacci numbers F_j^(r) are defined by (1-x-x^2)^{-r}=sum_{j>= 0}F_{j+1}^(r)x^j. In this note we consider some related numbers that can be expressed in terms of convolved Fibonacci numbers. These numbers appear in the numerical evaluation of a constant arising in the study of the average density of elements in a finite field having order congruent to a (mod d). We derive a formula expressing these numbers in terms of ordinary Fibonacci and Lucas numbers. The non-negativity of these numbers can be inferred from Witt's dimension formula for free Lie algebras. This note is a case study of the transform 1/n sum_{d|}n mu(d)f(z^d)^{n/d} (with f any formal series), which was introduced and studied in a companion paper by Moree.