The number $p_1(n)$ of overpartitions of $n$ is generated by $$\sum_{n\geq0}p_1(n)\,q^n=\prod_{k=1}^{\infty}\frac{1+q^k}{1-q^k}.$$ Let $t\in\mathbb{N}$. Now, extend this to construct a family of sequences $p_t(n)$ as generated by the product $$\sum_{n\geq0}p_t(n)\,q^n=\prod_{k=1}^{\infty}\left(\frac{1+q^k}{1-q^k}\right)^t.$$
I would like to ask:
QUESTION 1. For each $t$, is the sequence $p_t(n)$ log-concave in $n$, i.e. $p_t(n)^2\geq p_t(n+1)\,p_t(n-1)$ for all $n>1$?
This question is now answered by Gjergji Zaimi as shown below.
ADDED. More curiosity:
QUESTION 2. For each $n$, is the sequence $p_t(n)$ log-concave in $t$, i.e. $p_t(n)^2\geq p_{t+1}(n)\,p_{t-1}(n)$ for all $t>1$?