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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.

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    $\begingroup$ I asked for something similar here but no combinatorial answers so far. mathoverflow.net/questions/26440/… $\endgroup$
    – Gjergji Zaimi
    Commented Dec 19, 2016 at 7:48
  • $\begingroup$ @GjergjiZaimi: How did you come across your question? For me, it's because we wrote a paper on these numbers. $\endgroup$
    – T. Amdeberhan
    Commented Dec 19, 2016 at 15:15

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