Is the polynomial

$$P_n(x,y)=\displaystyle\sum_{a+b\leq n}x^ay^b$$

irreducible in $\mathbb Z[x,y]$?

For all $n\leq 500$ this is true (checked using Mathematica), so it is reasonable to presume that it is true for all $n$.

This question is related to another problem posted on this forum. Namely,

$$P_n(x,1)=\sum_{0\leq i\leq n} (n-i+1)x^i=f_n(x)$$

so it is easy to see that proving that $f_n(x)$ is irreducible should be enough. But this seems to be open...