Mathematica uses the same farmula as the cited book. With the difference that they omit the minus. The n and the k are subtrahated by 1 because they use another interval of summation.

Here is the Formular:

http://reference.wolfram.com/mathema...urier.en/1.gif
It can be found in the Wolfram Referece site, by pressing "more detail":

http://reference.wolfram.com/mathema...urier.en/1.gif
Cambridge "Numerical Recipes” uses as well the same formula as you but without minus.

EDIT:

The 1/sqrt(n) is just a scaling factor of minor relevance