I know, you have probably solved this Kata. Moreover there are already answers in comments, but anyway...
2**n this is the for the sum of binominal coefs, but there is another formula for sum of squares.

not an issue (issue = "for the kata"), just your approach that is incorrect. See here.

I am glad I looked at this comment.

22 being the thresholds makes me guess that you overflow on computation of factorial.
For some very general hints you can see here, especially third bullet point.

Hey guys. I am totaly stuck... I`ve got an algorithm to calculate the ansver in O(1). All basic tests are passed. But I constantly fail all tests, where the lenght of string equals 23,24,25. Some scary stuff happens, always like this:

'noabcdefgijklymzustvpxwrq' should equal 'nmzyxwvutsrqpolkjigfedcba'
'lzyxvutsrqonmeijhkcafdgb' should equal 'lzyxvutsrqonmkjihgfedcba'

I mean that it goes in the right direction, but suddenly mistakes. Give me a hint what I should check or what thing to fix. Thanks a lot!