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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",...
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