Fix a positive integer $k$. Then, the sequences $$c(n,k)=\frac{k^n}{n!}\prod_{m=1}^{n-1}(1+km)=[x^n]\left(\frac1{1-k^2x}\right)^{1/k}$$ are referred to as "$k$-central binomial coefficients", and these are always integers. For connection with differential Galois groups, see the paper.

QUESTIONS.What do the numbers $c(n,k)$ count? Or, can you provide some interpretation? Can you name some other connections?Even special cases $k=3, 4, \dots$ (and $n$ arbitrary) would be interesting.

**Example.** $c(n,2)=\binom{2n}n$ enumerate many familiar objects.