I don't think there is an exact analytical solution. Of course if you're only interested in the first d decimal points you can just rearrange the terms so the piece you're inserting the recurrence in is less than 10^-d for very large k (around what you'll be using). IIRC this is also the only way to solve problems in form x*e^(x) = k. The rearrange of terms is trivial in that case. I think it might be harder in this case.
