Apply [Lagrange reversion](https://en.m.wikipedia.org/wiki/Lagrange_reversion_theorem) 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](https://functions.wolfram.com/Polynomials/NorlundB2/)/[generalized Bernoulli numbers](https://www.fq.math.ca/Papers1/53-4/MollVignat04222015.pdf) appear and we get:

$$\boxed{c=\frac{k^3}{k^2-k+1},k=\sum_{m,n=0}^\infty\frac{B_m^{(\frac n2)}}{n! (m-n+1)!}}$$

[shown here](https://www.wolframalpha.com/input?i=evaluate+c%3Dk%5E3.0%2F%28k%5E2-k%2B1%29%2Ck%3Dsum%5Bsum%5BNorlundB%5Bm%2Cn%2F2.0%5D%2F%28n%21+%28m-n%2B1%29%21%29%2C%7Bm%2C0%2C20%7D%5D%2C%7Bn%2C0%2C20%7D%5D)