Apply Lagrange reversion to @TheSimpli’s equation:
$$\frac{k}{(k-1)^2}+1=e^k\iff k=1+\sqrt{\frac k{e^k-1}}\\\implies k=1+\sum_{n=1}^\infty\frac1{n!}\frac{d^{n-1}}{dx^{n-1}}\left(\frac x{e^x-1}\right)^\frac n2$$
Therefore Norlund polynomials/generalized Bernoulli numbers appear and we get:
$$\boxed{c=\frac{k^3}{k^2-k+1},k=\sum_{m,n=0}^\infty\frac{B_m^{(\frac n2)}}{n! (m-n+1)!}}$$