Degree-zero Segre and Verlinde coefficient formulas for rank one

Let $A_0(q),A_1(q),B_0(q),B_1(q)$ be the series introduced from the degree-zero reduced Segre and Verlinde series, with $N=1$. Degree-zero coefficient formulas. For $r\in\mathbb{Z}$,

$$[q^n]B_0(q)=\frac16\left(\binom{r+1}{2}(n-1)-1\right)[q^n]B_1(q).$$

When $r<-1$ and $n>1$,

$$[q^n]A_0(q)=\frac{1}{12}r(nr+n+2)[q^n]A_1(q).$$

The formulas are presented as conjectural and were checked in the source for $n\leq20$ and $r<5$; no proof or resolution is supplied.