Let $n$ be an odd number and consider the polynomial $$ X^{n-1}+X^{n-2}+\dots+X+\frac{n-1}{2n-1} $$ Is it possible to prove that this polynomial is irreducible in $\mathbb{Q}[X]$?
We can consider the polynomial $$ (n-1)X^{n-1}+(2n-1)X^{n-2}+\dots+(2n-1)X+(2n-1) $$ whose roots are the reciprocal of the roots of our polynomial. Since gcd$(n-1,2n-1)=1$ I can apply Eisenstein whenever $\exists p$ prime such that $p \mid 2n-1$ and $p^2 \nmid 2n-1$. But what about for example the case $n=5$?. Then $$ 4X^4+9X^3+\dots+9X+9 $$ and I cannot apply Eisenstein to show it's irreducible.
EDIT:(thanks to Sil and Dietrich Burde)
If $2n-1$ is a square and $n$ is prime, we can apply Eisenstein with $p=n$ to the shifted polynomial
$$ (2n-1)(X+1)^{n-1}+(2n-1)(X+1)^{n-2}+ \dots +(2n-1)(X+1)+(n-1) $$
Since the leading coefficient is $2n-1$, the constant term is $2n(n-1)$ and the $k$-th (with $1 \leq k \leq n-2$) coefficients are $(2n-1){{n}\choose{k+1}}$ which are divisible by $n$ since is prime.
So we are left with the case $n$ is composite and $2n-1$ is a square, i.e. the sequence A166080.
For $n=5$ the polynomial $$ 9X^4+9X^3+9X^2+9X+4 $$ is irreducible modulo $7$ and hence irreducible over $\Bbb Q$. Replacing $X:=Y+1$ (due to Sil) we obtain $$ 9Y^4 + 45Y^3 + 90Y^2 + 90Y + 40, $$ which is irreducible by Eisenstein with $p=5$.
The next case is $n=13$, where Eisenstein doesn't work directly for the reciprocal polynomial because $2n-1=5^2$. But then the polynomial is irreducible over $\Bbb F_{17}$ and $X:=y+1$ gives $$ 25y^{12} + 325y^{11} + 1950y^{10} + 7150y^9 + 17875y^8 + 32175y^7 + 42900y^6 + 42900y^5 + 32175y^4 + 17875y^3 + 7150y^2 + 1950y + 312, $$ which is Eisenstein with $p=13$. This is probably no coincidence. Shifted Eisenstein should work for many such numbers $n$ from OEIS $A001844$, see the remark.
Remark: The sequence of odd numbers $n$ such that $2n-1$ has no prime factor $p$ with $p\mid\mid 2n-1$ is the sequence A001844, starting with $$ 1, 5, 13, 25, 41, 61, 85, 113, 145, 181, 221, 265, 313, 365, 421, 481, 545, 613, 685, 761, 841, 925, \ldots $$