Log-Concavity of Sequences Related to Overpartitions

Question

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 $t>1$?

Answer 1

This follows from the fact that $p_1(n)$ is a log-concave sequence, together with the fact that the convolution of log concave sequences is also a log concave sequence.

The first fact is proven in B. Engel "Log-concavity of the overpartition function", Ramanujan J 43, 229–241 (2017).

The second is proven in S. Hoggar "Chromatic polynomials and logarithmic concavity", J. Combin. Theory Ser. B 16 (1974) 248–254.