An interesting divisibility about a Dirchlet convolution of binomial coefficients with Mobius function I have found some interesting divisibility properties which I don't know how to prove. 
If we set $$T(n,k)=\sum_{l|n}\mu(l)(-1)^{\frac{n}{l}}\binom{k\frac{n}{l}}{\frac{n}{l}}$$ where $\mu(.)$ is the Mobius function, then $T(n,2k)$ is probably divisible by $2kn^2$, and $2T(n,2k+1)$ is divisible by $(2k+1)n^2$.  
Some verifications: 
It's not difficult to show that if $n=p$ is a prime, then by Wilson's theorem, we have $T(p,k)$ divisible by $pk$, but are they divisible by $p^2k$?
Some numerical verifications:
$T(1,1)=-1$ and $T(2,1)=2$ and $T(n,1)=0$ for all $n\geq 3$.
$T(2,k)=2k^2$ is divisible by $4k$ (or $2k$) when $k$ is even (or odd).
$18|T(3,2)=-18$, $32|T(4,2)=64$, $50|T(5,2)=-250$, $72|T(6,2)=936$, $98|T(7,2)=-3430$
$81|T(3,3)=-81$, $48|T(4,3)=480$, $75|T(5,3)=-3000$, $54|T(6,3)=18564+84-15-3=18630$.

I have found the following paper:
https://arxiv.org/abs/1504.06327
Proposition 1.2. gives the integrality we want since these invariants are integers! But I am still waiting for a direct proof. For exemple, the paper here 
https://arxiv.org/abs/1703.00990
gives a partial proof. 
