The number $p\_1(n)$ of [overpartitions of $n$](https://oeis.org/A015128) 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:



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> **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 $n>1$?
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This question is now answered by Gjergji Zaimi as shown below.


**ADDED.** More curiosity:



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> **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 $t>1$?
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Log-concavity of sequence related to overpartitions
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.


