My code

… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too. Or just jump to my GitHub repository.

       
#include <iostream>
       
#include <iomanip>
       
#include <vector>
       
#include <algorithm>
       
 
       
// execute all moves on all permutations of a given size
       
// → only needed to get an idea what the formula looks like, not needed anymore

       
double evaluate(unsigned int size)
       
{
       
  // data = { 1,2,3,...,size }
       
  std::vector<unsigned int> data, current;
       
  for (unsigned int i = 1; i <= size; i++)
       
    data.push_back(i);
       
 
       
  // count moves and permutations
       
  unsigned int moves = 0;
       
  unsigned int permutations = 0;
       
  do
       
  {
       
    // get current permutation
       
    current = data;
       
    permutations++;
       
 
       
    // sort container
       
    unsigned int pos = 1;
       
    // compare data[pos - 1] and data[pos]
       
    while (pos < size)
       
    {
       
      // pair in wrong order ?
       
      if (current[pos] < current[pos - 1])
       
      {
       
        // rotate all elements from 0 to pos to the right once (last element becomes the first)
       
        std::rotate(current.begin(), current.begin() + pos, current.begin() + pos + 1);
       
        moves++;
       
        // restart
       
        pos = 1;
       
      }
       
      else
       
        pos++;
       
    }
       
  } while (std::next_permutation(data.begin(), data.end()));
       
 
       
  std::cout << "E(" << size << ")=" << moves / double(permutations) // result
       
            << " =" << moves << "/" << permutations << std::endl;   // and as a fraction
       
  return moves / double(permutations);
       
}
       
 
       
int main()
       
{
       
  unsigned int limit = 10;
       
  std::cin >> limit;
       
 
       
  std::cout << std::fixed << std::setprecision(2);
       
 
       
  // solve for small input values
       
  //for (unsigned int i = 1; i <= limit; i++)
       
  //  evaluate(i);
       
 
       
  double result = 0;
       
  for (unsigned int i = 1; i <= limit; i++)
       
  {
       
    // 2^(i-1) - 1
       
    auto numerator = (1 << (i - 1)) - 1; // 0,1,3,7,15,31,...
       
    // add to result
       
    result += numerator / double(i);
       
  }
       
  std::cout << result << std::endl;
       
  return 0;
       
}

This solution contains 8 empty lines, 15 comments and 4 preprocessor commands.