# in search of intepretations and connections for $k$-central binomials

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.

• I asked for something similar here but no combinatorial answers so far. mathoverflow.net/questions/26440/… – Gjergji Zaimi Dec 19 '16 at 7:48
• @GjergjiZaimi: How did you come across your question? For me, it's because we wrote a paper on these numbers. – T. Amdeberhan Dec 19 '16 at 15:15