It is well known that the complete homogeneous symmetric polynomial $h_{n-k}(1,\,2,\,3, ...,\,k-1,\,k)$ equals $S(n,\,k)$ the Stirling number of the second kind. [Wikipedia]

During a research project I stumbled upon the following complete homogeneous symmetric polynomial: $h_{n-k}(1,\,2,\,3, ...,\,k-1,\,n)$.

My question is: is the latter symmetric polynomial expressible in nice, simple and/or interpretable terms?

Or is this too much to ask? If so, why?