I computed the Galois group of $$P_n := x^n + 2x^{n-1} + \dots + nx - 1$$ for $n=2,3,4,5,7$ with the help of Dedekind's theorem, and each time it was $S_n$. So it made me wonder: is it always $S_n$? In fact, I'm not even sure if these polynomials are all irreducible over the rationals. We probably cannot use Eisenstein's criterion for all of them, nor Perron's. There was no clear pattern in the decompositions of these polynomials modulo some primes, so I'm not sure how to attack this question.
These polynomials appeared to me in the following context: let say that at step $0$ we have a population of one person, and at each step, each person of the population independently has $0,1,2,...$ or $n$ kids, and then dies, and each possibility happens with probability $\frac{1}{n+1}$. Then, the probability that the population will die at some point is the smallest real positive root of $P_{n-1}$. If the Galois group is $S_{n-1}$, then for $n > 5$, the probability is not expressible with radicals.
Edit: using Magma, I verified the claim for $n \leq 100$. I also verified that $P_n$ was irreducible for $n \leq 1100$.
