Rational sums of inverse powers of fixed points of e

[Gottfried]

This post doesn't have much if anything to do with tetration, but I discovered it while analyzing logarithms at the fixed points. I had mentioned this previously:
http://math.eretrandre.org/tetrationforu...php?tid=59

The fixed points of e are irrational. However, the sums of the inverse n-th powers are rational.

Hmm, how did you compute this actually?
I tried it with Henryks fixed-point-formula, getting the reciprocals of the first few fixpoints of e

Code:

.
        0.168376379087-0.707754188785*I // branch 0
       0.0333484363677-0.122713337947*I // branch 1
      0.0131593793714-0.0691856926303*I
     0.00718943124116-0.0482569114994*I
     0.00458333618210-0.0370546869791*I
     0.00319972064661-0.0300725070538*I

Does this agree with your data?

Gottfried