Theorems and conjectures on some rational generating functions

Let $F_i$ denote the $i$th Fibonacci number, and define ${\prod }_{i=1}^n\left(1+\right)\left(x^{F_{i+1}}\right)={\sum }_k{c}_n\left(k\right)x^k$. The paper is concerned primarily with the coefficients $c_n\left(k\right)$. In particular, for any $r\ge 0$ the generating function ${\sum }_{n\ge 0}\left({\sum }_k{c}_n{\left(k\right)}^r\right)x^n$ is rational. The coefficients $c_n\left(k\right)$ can be displayed in an array called the Fibonacci triangle poset $F$ 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.