From: jos.bergervoet@nxp.com   
      
   On 19/06/03 3:35 AM, Jos Bergervoet wrote:   
   > On 19/05/25 4:06 PM, MM wrote:   
      ...   
   >>>> How to find out the series expansion for s = s(t) given by the implicit   
   >>>> equation:   
   >>>>   
   >>>> 18 t + s(1+s) (1+2s) =0   
   >>>>   
   >>>> There actually exists a closed expression for series s(t) = sum_k s_k   
   >>>> t^k   
       ...   
   > You could start in the usual way with Mathematica:   
     ...> ser = Series[s2, {t,0, 7}]   
   >   
   > (* Anyhow, you can now ask for the series coefficients:  *)   
   >   
   > SeriesCoefficient[ser, 0]   
   > SeriesCoefficient[ser, 1]   
   >      (* ... *)   
   > SeriesCoefficient[ser, 6]   
   > SeriesCoefficient[ser, 7]   
   >   
   >> I don't see any obvious way to find out the generic s_k .   
   >   
   > If you want a closed form expression for any k then the above   
   > approach does not immediately help..   
      
   Maybe interesting for those who like puzzles. The expansion   
   coefficients around t=0 for powers t^n from 0 to 12 are:   
      
   -1   
   -18   
   972   
   -93312   
   11022480   
   -1451188224   
   204277417344   
   -30091839012864   
   4581008631403776   
   -714982094945648640   
   113792254404069795840   
   -18397555276256731201536   
      
   Can you see the expression in n? The actual function s(t) being   
   expanded is: (see previous post)   
      
                1                     1 + I Sqrt[3]   
       s2   = -(-) - ------------------------------------------------ -   
                2       1/3                                    2  1/3   
                     4 3    (-324 t + Sqrt[3] Sqrt[-1 + 34992 t ])   
      
                                                            2  1/3   
         (1 - I Sqrt[3]) (-324 t + Sqrt[3] Sqrt[-1 + 34992 t ])   
         ---------------------------------------------------------   
                                     2/3   
                                  4 3   
      
   --   
   Jos   
      
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