lu := - (

                                                                             3
    x[1, 1] x[1, 2] x[1, 3] x[2, 1] x[2, 2] x[2, 3] x[3, 1] x[3, 2] x[3, 3] z

     - 1)/((x[2, 2] x[1, 3] z x[3, 1] - 1) (x[2, 3] x[1, 2] z x[3, 1] - 1)

    (x[2, 1] x[1, 2] z x[3, 3] - 1) (x[2, 2] x[1, 1] z x[3, 3] - 1)

    (x[2, 3] x[1, 1] z x[3, 2] - 1) (x[2, 1] x[1, 3] z x[3, 2] - 1))

      The weighted generating function for magic squares of size, 3, is:

                                                                            3
- (x[1, 1] x[1, 2] x[1, 3] x[2, 1] x[2, 2] x[2, 3] x[3, 1] x[3, 2] x[3, 3] z

     - 1)/((x[2, 2] x[1, 3] z x[3, 1] - 1) (x[2, 3] x[1, 2] z x[3, 1] - 1)

    (x[2, 1] x[1, 2] z x[3, 3] - 1) (x[2, 2] x[1, 1] z x[3, 3] - 1)

    (x[2, 3] x[1, 1] z x[3, 2] - 1) (x[2, 1] x[1, 3] z x[3, 2] - 1))

                          and in Maple input notation:

-(x[1,1]*x[1,2]*x[1,3]*x[2,1]*x[2,2]*x[2,3]*x[3,1]*x[3,2]*x[3,3]*z^3-1)/(x[2,2]
*x[1,3]*z*x[3,1]-1)/(x[2,3]*x[1,2]*z*x[3,1]-1)/(x[2,1]*x[1,2]*z*x[3,3]-1)/(x[2,
2]*x[1,1]*z*x[3,3]-1)/(x[2,3]*x[1,1]*z*x[3,2]-1)/(x[2,1]*x[1,3]*z*x[3,2]-1)
                                        2
                                       z  + z + 1
                               lu := - ----------
                                               5
                                        (z - 1)

           The generating function for magic squares of size, 3, is:

                                     2
                                    z  + z + 1
                                  - ----------
                                            5
                                     (z - 1)

                          and in Maple input notation:

-(z^2+z+1)/(z-1)^5
                           And the squence itself is:

                          4        3         2
                   [[1/8 n  + 3/4 n  + 15/8 n  + 9/4 n + 1]]

                          and in Maple input notation:

[[1/8*n^4+3/4*n^3+15/8*n^2+9/4*n+1]]
                         This took, 6540.671, seconds.