Simple whole-number convolution vectors

Smoothing:
[1 1 1]         = 3 point boxcar (sliding average) smooth
[1 1 1 1]       = 4 point boxcar (sliding average)smooth
[1 2 1]         = 3 point triangular smooth
[1 2 3 2 1]     = 5 point triangular smooth
[1 4 6 4 1]     = 5 point Gaussian smooth  
[1 4 8 10 8 4 1]       = 7 point Gaussian smooth  
[1 4 9 14 17 14 9 4 1] = 9 point Gaussian smooth  

Differentiation:
[-1 1]         First derivative
[1 -2 1]       Second derivative
[1 -2 1 -1]    Third derivative
[1 -4 6 -4 1]  Fourth derivative

Results of successive convolution by two vectors:
Conv 1		Conv 2						
[1 1 1]     x   [1 1 1]	 = [1 2 3 2 1]	Triangular smooth			
[1 2 1]     x 	[1 2 1]	 = [1 4 6 4 1]	Pesudo-Gaussian smooth			
[-1 1]      x 	[-1 1]	 = [1 -2 1]	2nd derivative			
[-1 1]      x 	[1 -2 1] = [1 -3 3 -1]	3rd derivative			
[-1 1]      x 	[1 1 1]  = [1 0 0 -1)	1st derivative with gap-segment			
[-1 1]      x 	[1 2 1]	 = [1 1 -1 -1)	Smoothed 1st derivative			
[1 1 -1 -1] x	[1 2 1]	 = [1 3 2 -2 -3 -1]	1st derivative with more smoothing		
[1 -2 1]    x	[1 2 1]	 = [1 0 -2 0 1]	2nd derivative with gap-segment			
rectangle   x  rectangle = triangle or trapezoid
Gaussian  x  Gaussian    = Gaussian of greater width
Gaussian x  Lorentzian   = Something in between Gaussian and Lorentzian