I found this example on wikipedia. What I don't understand is that on the right hand side, the column starts with $-10$. Why isn't the column $-10, -4, 4, 10$? Why do the numbers in the middle place need to be divided by 2? Thanks!
Edit: Also why is the third number of the column starts with 15 1? $1-(-1)=2$?
The recursion formula for divided differences is $$ f[x_n, \ldots, x_m] = \frac{f[x_n, \ldots, x_{m+1}] - f[x_{n-1},\ldots, x_m]}{x_n- x_m} $$ In the cases you mention, the differences in the denominator is $2$, so the difference is divided by $2$.