Geodesic
Convoluted convolved Fibonacci numbers
Journal of integer sequences, Tome 7 (2004) no. 2.

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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.
Keywords: circular words, monotonicity, Witt's dimension formula, Dirichlet L-series 
@article{JIS_2004__7_2_a5,
     author = {Moree, Pieter},
     title = {Convoluted convolved {Fibonacci} numbers},
     journal = {Journal of integer sequences},
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     volume = {7},
     number = {2},
     year = {2004},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2004__7_2_a5/}
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Moree, Pieter. Convoluted convolved Fibonacci numbers. Journal of integer sequences, Tome 7 (2004) no. 2. http://geodesic.mathdoc.fr/item/JIS_2004__7_2_a5/