How to determine the filter type of a discrete frequency spectrum
The $k_1$ indexes you get from the FFT correspond to frequencies 0 to $2\pi$ (actually, 1 sample short of $2\pi$). Note that frequencies in the range $\pi$ to $2\pi$ also correspond to the negative half of the specturm ($-\pi$ to 0).

You can get a nice visualisation with frequency 0 in the middle by applying fftshift to the result (Octave or Matlab assumed).

You can also check freqz, which plots a nice frequency and phase repsonse (try freqz([1 2 1], [1])). Or try plot(abs(freqz([1 2 1],[1]))) to get a linear plot rather than log.

What you are getting is a low-pass filter (gain 4 at low frequency, 0 at high frequency).

In VERY general terms: You can know beforehand it is a LPF because the impulse response is a single bump. A high-pass filter will have some sort of antisymetric shape (like [1 2 -2 -1]), and a band-pass or band-reject will have some oscillations in the impulse response.