Theorems and conjectures on some rational generating functions

Abstract

Let Fi denote the ith Fibonacci number, and define ∏i=1n1+xFi+1=∑kcn(k)xk. The paper is concerned primarily with the coefficients cn(k). In particular, for any r≥0 the generating function ∑n≥0(∑kcn(k)r)xn is rational. The coefficients cn(k) can be displayed in an array called the Fibonacci triangle posetF with some interesting further properties, including an encoding of a certain dense linear order on the nonnegative integers. Some generalizations are briefly considered, but there remain many open questions.