Let $p(n)$ denote the number of (unrestricted) integer partition of $n$ which has the product generating function $$\sum_{n\geq0}p(n)\,x^n=\prod_{j\geq1}\frac1{1-x^j}.$$ On the other hand, for the number of partitions of $n$ with $k$ parts, $p(n, k)$, we may use a two-variable generating function $$\sum_{n,k\geq0}p(n,k)\,x^nt^k=\prod_{j\geq1}\frac1{1-tx^j}.$$ Define the polynomials $Q_n(t)=\sum_{k\geq0}p(n,k)\,t^k$. Here are the first few such polynomials, for $1\leq n\leq 11$, \begin{align*} Q_1(t)&=t \\ Q_2(t)&=t(t+1) \\ Q_3(t)&=t(t^2+t+1) \\ Q_4(t)&=t(t^3+t^2+2t+1) \\ Q_5(t)&=t(t^4+t^3+2t^2+2t+1) \\ Q_6(t)&=t(t^5+t^4+2t^3+3t^2+3t+1) \\ Q_7(t)&=t\mathbf{(t^2+t+1)(t^4+t^2+2t+1)} \\ Q_8(t)&=t(t^7+t^6+2t^5+3t^4+5t^3+5t^2+4t+1) \\ Q_9(t)&=t(t^8+t^7+2t^6+3t^5+4t^4+6t^3+7t^2+4t+1) \\ Q_{10}(t)&=t\mathbf{(t^2+t+1)(t^7+t^5+2t^4+2t^3+3t^2+4t+1)}. \end{align*} My intent here is to focus on the "non-trivial" factorizations that appear in $Q_7(t)$ and $Q_{10}(t)$. Hence, I ask:
QUESTION. Are there finitely many or infinitely many non-trivially factorizable polynomials $Q_n(t)$, say over $\mathbb{N}[t]$?
