This comes in relation to the Fishburn numbers.
I stumbled on the following relation for which I ask a proof if true.
Let $Q_i(z):=1-(1-t)^{i-1}(1-zt)$. Then $$\sum_{n=0}^{\infty}\frac{(n+1)zt}{(1-zt)^{n+1}}\prod_{i=1}^nQ_i(1)= \sum_{m=2}^{\infty}\sum_{j=2}^m\prod_{i=1,\, i\neq j}^mQ_i(z) - \sum_{m=2}^{\infty}(m-1)\prod_{i=1}^mQ_i(z).$$
