Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.2

Convoluted Convolved Fibonacci Numbers

Pieter Moree
Max-Planck-Institut für Mathematik
Vivatsgasse 7
D-53111 Bonn

Abstract: 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.

This note is a case study of the transform ${1\over n}\sum_{d {\, \vert \,}n}\mu(d)f(z^d)^{n/d}$ (with $f$ any formal series), which was introduced and studied in a companion paper by Moree.

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(Concerned with sequences A000096 A006504 A001628 A001870 A001629 .)

Received November 12 2003; revised version received April 20 2004. Published in Journal of Integer Sequences, April 26 2004.

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