It seemed like a neat idea, but through 5th order, at least, it coincides with PadeApproximant[f[y],{y,0,{<n or n+1>,n}}]. Coalescing the points of the 5th reciprocal difference took 15 cpu min to get 0/0, which then took 14 l'Hospitalings ("A very large output was generated", 3 min each) to get (3 (5 (f^(4))[0]^2-4 (f^(3))[0] (f^(5))[0]))/(40 (f^(3))[0]^3-60 (f^\[Prime]\[Prime])[0] (f^(3))[0] (f^(4))[0] +15 (f^\[Prime])[0] (f^(4))[0]^2+18 (f^\[Prime]\[Prime])[0]^2 (f^(5))[0]-12 (f^\[Prime])[0] (f^(3))[0] (f^(5))[0]) The required l'Hospitalings (l'Hospitalizations?) went 0,1,2,5,10,14,... --rwg (with Corey and Julian)