@Gokou: The last term which you are taking as a constant in equation (1) is not actually a constant.
It's 2x1 + 22x2 + 23x3 + ... + 2k-1x(k-1)
So if you solve this series, you should get the correct answer. I remember solving this way a little while ago, see if you can solve it and post the solution, please.
Edit: I think this way of solving recurrence is not correct because we're assuming the input size to be a power of 2. We must also prove that the recurrence holds for other input sizes.
Edit2: The sum of this series would be n(lg n - 2) + 2.
So T(n) = 2k T(n/2k) + lg n (n - 1) - [ n (lg n - 2) + 2]
Solving it you'll get T(n) = cn - lg n - 2