tl;dr
Can Savitzky-Golay finite difference coefficients be derived from Pascal's triangle, or from combinations of Pascal's triangles?
More detail
Here is Pascal's triangle:
It has several useful number patterns in it link including finite difference.
In calculus, finite difference approximations turn symbolic partial differential equations into algebraic relationships that are solvable with computers.
Here is a list of some first order forward finite differences:
\begin{matrix}
& f_i & f_{i+1} & f_{i+2} & f_{i+3} & f_{i+4} \\
h\cdot{f}'\left( x_i \right ) & -1 & +1 & 0 & 0 & 0 \\
h^2\cdot{f}''\left( x_i \right ) & +1 & -2 & +1 & 0 & 0 \\
h^3\cdot{f}'''\left( x_i \right ) & -1 & +3 & -3 & +1& 0\\
h^4\cdot{f}^{iv}\left( x_i \right ) & +1 & -4 & +6 & -4 & +1
\end{matrix}
Inspection shows that this is found by imposing a pattern of signs to the rows of Pascal's triangle. You can see that the first row of this table is "-1 +1" and the second row of Pascal's triangle is "1 1". The second row of the table has magnitudes "1 -2 1" and the second row of the pyramid is "1 2 1". There is a clean pattern in the signs.
Savitzky-Golay coefficients are a robust finite difference. It is like the above finite difference, but where the above can be strongly moved from "actual" by the addition of small noises, this is much more resistant. They are used in spectroscopy, chemistry, and actuarial science.
Here are some Savitzky-Golay coefficients: link.
While these are centered (I didn't find a table of forward ones), there exist both forward and backward Savitzky-Golay coefficients.
For conventional finite differences, sequential forward differences can be combined to make a centered difference expression. The average of the 2-point first order forward finite difference, gives a 3 point centered first order finite difference.
Is there a pattern for deriving Savitzky-Golay coefficients from Pascal's triangle?