Recall the [integer partition function](https://oeis.org/A000041) $P(n)$ with generating function
$$\sum\_{n\geq0}P(n)x^n=\prod\_{k=1}^{\infty}\frac1{1-x^k}.$$
Let $[n]\_q=\frac{1-q^n}{1-q}$ denote the $q$-analogue of the integer $n$ and let $\lambda=(\lambda\_1,\lambda\_2,\dots)\vdash n$ be a partition of $n$. Now, define the function
$$\Psi\_q(n)=\sum\_{\lambda\vdash n}\,\,\sum\_{j\geq1}\,\, [\lambda\_j]\_q.$$
For example, $\Psi\_q(3)=[3]\_q+([2]\_q+[1]\_q)+([1]\_q+[1]\_q+[1]\_q)=q^2+2q+6$. In particular, when $q=1$, we obtain $\Psi\_q(n)=nP(n)$ with [generating function](https://oeis.org/A066186)
$$\sum\_{n\geq1}nP(n)x^n=x\frac{d}{dx}\prod\_{k=1}^{\infty}\frac1{1-x^k}.$$


I would like to ask



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> **Question 1.** Is there a generating function for the sequence of $q$-polynomials $\Psi\_q(n)$?
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> **Question 2.** What is the combinatorial interpretation of the coefficients of $\Psi\_q(n)$?
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For example, the constant term of $\Psi\_q(n)$ enumerates [the number of $1$'s](https://oeis.org/A006128) in all partitions of $n$.



Partitions, $q$-polynomials and generating functions
A generating function of sorts is given by
 $$ \sum\_{n\geq 1}\Psi\_q(n)x^n = P(x)\sum\_{m\geq 0}q^m\sum\_{k\geq m+1}
 \frac{x^k}{1-x^k}, $$
where $P(x)=\prod\_{i\geq 1}(1-x^i)^{-1}$.


