Not using a negative exp in the definition of the forward DFT is a really bad idea. I know that *Numerical Recipes* uses a positive exp, but it's still a bad idea. I fear that most uses of the positive exp can be traced to be influenced by *Numerical Recipes* (*).

Whether you include the 1/N term in the forward or backward transform or not at all is really just a convention. There are at least two good reasons to not include the 1/N term:

- The FFTW library is a de facto reference implementation of the FFT, and it doesn't include the 1/N term.
- It would be unclear whether it should be included in the forward or backward transform, and hence a constant source of confusion.

Another reason in favor of not including the 1/N term is that it has become common practice (**) to define the (continuous) Fourier transform in a way that no normalization factors are needed (by using the frequency $\xi$ instead of the angular frequency $\omega$):

$\hat{f}(\xi) = \int_{-\infty}^{\infty} f(x)\ e^{- 2\pi i x \xi}\,dx$

$f(x) = \int_{-\infty}^{\infty} \hat{f}(\xi)\ e^{2 \pi i x \xi}\,d\xi$

So omitting the 1/N term from the DFT makes the definition look more similar to the continuous case. However, I have to admit that I have the habit of including the 1/N factor in the forward transform, because then the DFT of a constant function is independent of N.

(*)
You probably don't believe me ("It's just a convention, why should it matter?"), but I recently googled for NFFT (nonequispaced FFT) and read some of the related papers. One of the authors used the positive exp in his thesis, but changed to the negative exp in later papers and presentations. While reading one of his later presentations, I was surprised by the amount of sign mistakes and surprising differences in sign conventions between closely related definitions.

(**) I think this change must have occurred only in the last few years. Wikipedia claims that the mathematics literature always used the "frequency convention" and only the physics literature used the "angular frequency convention", but this doesn't match with my own experience.