[seqfan] Re: Self-avoiding walks on nXnXn cubic lattice, guess the coefficients

[Simon Plouffe]

Hello,

  about this series of polynomials,

  could it be that there is a gen. function
for the whole bunch of polynomials at once ??

  it does work with GFUN and a large class
of gen. functions like the orthogonal polynomials
for example.

  here is an example ;

  If T(n,x) is the n'th Chebychev polynomial then
we construct this series of polynomials in the
variable v.

            2          2    3     3
[1, v x, v  (-1 + 2 x ), v  (4 x  - 3 x),

      4         4      2    5      5       3
     v  (1 + 8 x  - 8 x ), v  (16 x  - 20 x  + 5 x),

      6           6       4       2
     v  (-1 + 32 x  - 48 x  + 18 x ),

      7      7        5       3
     v  (64 x  - 112 x  + 56 x  - 7 x),

      8           8        6        4       2
     v  (1 + 128 x  - 256 x  + 160 x  - 32 x ),

      9       9        7        5        3
     v  (256 x  - 576 x  + 432 x  - 120 x  + 9 x),

      10            10         8         6        4       2
     v   (-1 + 512 x   - 1280 x  + 1120 x  - 400 x  + 50 x )

        11
     , v   (

           11         9         7         5        3
     1024 x   - 2816 x  + 2816 x  - 1232 x  + 220 x  - 11 x)

        12            12         10         8         6
     , v   (1 + 2048 x   - 6144 x   + 6912 x  - 3584 x

             4       2
      + 840 x  - 72 x )]

  Then we ask : guessgf(%,v);

 > guessgf(%,v);
                            2
                         x v  - 1
                  [- ----------------, ogf]
                           2    4
                     -2 x v  + v  + 1

  That first operand is the gen. function of
the Chebychev polynomials, indeed if I expand
in terms of V (not x) then
we get :

        2            2   4       3         6
1 + x v  + (-1 + 2 x ) v  + (4 x  - 3 x) v  +

             4      2   8        5       3         10
     (1 + 8 x  - 8 x ) v  + (16 x  - 20 x  + 5 x) v   +

               6       4       2   12
     (-1 + 32 x  - 48 x  + 18 x ) v   +

          7        5       3         14
     (64 x  - 112 x  + 56 x  - 7 x) v   +

               8        6        4       2   16
     (1 + 128 x  - 256 x  + 160 x  - 32 x ) v   +

           9        7        5        3         18
     (256 x  - 576 x  + 432 x  - 120 x  + 9 x) v   +

                10         8         6        4       2   20
     (-1 + 512 x   - 1280 x  + 1120 x  - 400 x  + 50 x ) v

      + (

           11         9         7         5        3
     1024 x   - 2816 x  + 2816 x  - 1232 x  + 220 x  - 11 x)

      22      24
     v   + O(v  )

All of them, (which validates that we were not wrong).

  This trick can be applied to a LOT of tableaux of
coefficients in 2 variables.

  Ref :
http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.em/1048610118

Best regards,

  Simon Plouffe