TITLE: Looking for finite difference approximations past the fourth derivative
QUESTION [6 upvotes]: I scanned the internet and could not find further representations of the central difference approximations past the fourth derivative. Are there published results past the fourth derivative?
Ideally I am looking to find up to the $12$th derivative. It doesn't sound like an enjoyable exercise by pencil and paper. Is this information known?
Central difference approximations up to the $4$th derivative:
$f'(x)\approx\dfrac{-f(x+2h)+8f(x+h)-8f(x-h)+f(x-2h)}{12h}$
$f''(x)\approx\dfrac{-f(x+2h)+16f(x+h)-30f(x)+16f(x-h)-f(x-2h)}{12h^2}$
$f^{(3)}(x)\approx\dfrac{f(x+2h)-2f(x+h)+2f(x-h)-f(x-2h)}{2h^3}$
$f^{(4)}(x)\approx\dfrac{f(x+2h)-4f(x+h)+6f(x)-4f(x-h)+f(x-2h)}{h^4}$

REPLY [4 votes]: Here's what I've found on the topic
Notes on the derivation of Finite Differences kernels, on regularly spaced grids, using arbitrary sample points
Some tables on Finite differences can be found here on wiki
There is a paper that discusses the generation of the co-efficients used 
Generation of Finite Difference Formulas on Arbitrarily Spaced Grids
The interesting thing about this paper is that the gird can be arbitrarily spaced. Note that this paper presents an algorithm for computing the co-efficients. 
I've found this
$\delta^n_h[f](x)=\Sigma_{i=0}^{n} (-1)^iC_i^nf(x+(\frac{n}{2} -i)h)$
on wikipedia although I'm still searching for a derivation.