2 problems

Least-denominator conjecture. $\bar{C}(n,k)$ is a rational function with least denominator equal to the denominator of the $(n-1)(k-1)$-st derivative of $F(t)$.

Let $G(t)$ be the rational function … whose extra numerator factor is $(1+t+\cdots+t^{n-1})^2$. Equivalently, if $G(t)=\prod_i(1-t^i)^{-\beta(i)}$, then … Here $\bar{R}(n,k)$ denot…