Summary: The convolved Fibonacci numbers $F_j^{(r)}$ are defined by $(1-x-x^2)^{-r}=\sum_{j\ge 0}F_{j+1}^{(r)}x^j$. 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.

circular words, monotonicity, Witt's dimension formula, Dirichlet L-series