Given a number $k$, is it possible to find which term in the $n$-Simplex sequence it corresponds to? I've only been able to find formulae for the triangular root.
Examples:
For $k = 10$ in the triangular (2-Simplex) number sequence, the triangular root would be 4, since the fourth triangular number is 10.
For $k = 210$ in the polytope (4-Simplex) number sequence, the pentatonic root would be 7, since the seventh pentatope number is 210.
In order to do this in general, you would have to find the inverse function of the map $$ n \mapsto \binom{n+k-1}{k} $$ for fixed $k$. For small $k$, you got a nice little polynomial, and you may succeed. For large $k$, if you succeed, you've done better than these guys:
https://mathoverflow.net/questions/143332/is-there-an-asymptotic-formula-for-an-inverse-function-of-the-binomial-coefficie