4 problems

Positive-coefficient numerator conjecture. One can write

Let $A_{n,k}$ be the ballot numbers … For all $n,r_1,\ldots,r_m\in\mathbb{Z}^+$ such that $r_1+\cdots+r_m\equiv1\pmod2$, and with $\varepsilon=\pm1$, Guo–Zeng's general ballot-prod…

Let $A_{n,k}$ be the ballot numbers … For all $n,r,s,t\in\mathbb{Z}^+$ with $r+s+t\equiv1\pmod2$, and with $\varepsilon=\pm1$, Guo–Zeng's three-factor ballot-number conjecture. The…

Let $A_{n,k}$ be the ballot numbers … For $r\in\mathbb{N}$, $s,t\in\mathbb{Z}^+$ with $r+s+t\equiv1\pmod2$, $n\in\mathbb{Z}^+$, and $\varepsilon=\pm1$, Guo–Zeng's adjacent-index co…