Both questions are answered in the 2011 <A HREF="https://ir.lib.uwo.ca/cgi/viewcontent.cgi?article=7090&context=digitizedtheses">Ph.D. thesis</A> of G.A. Kalugin, see also <A HREF="https://arxiv.org/abs/1011.5940">arXiv:1011.5940</A>

$$\alpha_{n,k}=\sum_{m=0}^k\frac{1}{m!}{{2n-1}\choose{k-m}}\sum_{q=0}^m(-1)^q (q+n)^{m+n-1}.$$
There are equivalent representations in terms of Stirling numbers or Bernoulli polynomials.    

In section 4.2.3 Kalugin proves that the coefficients are positive, unimodal, and log-concave.