Generalized distribution time series moving averages I'm trying to figure out a pattern that I'm seeing. I have weights generated by an unknown function F
F(3) = [-3/9, 2/9, 6/9, 3/9, 1/9]
F(4) = [-8/16, -1/16, 5/16, 10/16, 6/17, 3/16, 1/16]
F(5) = [-15/25, -6/25, 2/25, 9/25, 15/25, 10/25, 6/25, 3/25, 1/25]
...

I understand how the function is generating the series/weights, but I'm wondering if there's any more nuance or theory behind the weight generation. I know that that weights are generated according to some time series manipulation (likely manipulations of moving averages of different lengths). Curious if anyone has seen anything like this and could enlighten me on the theory behind these weights
 A: Looks to me like they are the coefficients for a Finite Impulse Response filter. Note how the sum of the numerators == denominator for each given example.
A: These are coefficients / weights of a type of univariate time series filter, commonly known as a "corrective moving average": the negative weights at the back of the window indicate that it was designed to minimize "moving average lag". It is indeed a FIR filter, as mentioned above. If you want to learn about lag, I'd suggest you read Ehlers' work, and learn about the difference between group delay and phase delay.
Well-known examples of this class of filters are
ZLEMA by John Ehlers, T3 moving average introduced by Tim Tillson, etc. (these are not FIR filters), or Hull moving average by Alan Hull (FIR: if based on a WMA).
So to quote the OP: "likely manipulations of moving averages of different lengths", is correct.
It does get interesting if you can turn this static weight function into a dynamic / adaptive filter, by testing the effect of different shapes of the weights on the phase shift with a sinusoidal signal.
It would help if you shared the function that generated these.
