Theorem: Let M(n) be the number of walks of length n

             from (0,0,0) using unit positive steps and staying in

          the region x>=y>=z>=0. Let A(n) be the number of such walks

                  with n-3 steps, that start at (2,1,0). Then

                             A(n)=M(n-1)-M(n-3) .



                                     Proof:

                             The summand for M(n),

                            let's call it F(n,k), is

                                       n!
                            ------------------------
                                        2
                            (k + 1) (k!)  (n - 2 k)!

                             The summand for A(n),

                            let's call it G(n,k), is

                        2                      2
                    2 (n  - 2 n - 2 n k - 1 + k  + 4 k) k n!
                   ------------------------------------------
                                                 2
                   (k + 1) n (n - 1) (n - 2) (k!)  (n - 2 k)!

              To prove Amitai Regev's conjecture we must show that

               Sum(F(n-1,k)-F(n-3,k),k=0..n)=Sum(G(n,k),k=0..n).

                      Hence the summand of the difference,

                         F(n-1,k)-F(n-3,k)-G(n,k), is

                          2                            2
                    2 k (n  - n + 1 - 4 n k + 2 k + 3 k ) n!
                   ------------------------------------------
                                                 2
                   n (n - 1) (n - 2) (k + 1) (k!)  (n - 2 k)!

        But this is a telescoping sum, the summand being H(n,k+1)-H(n,k)

                                where H(n,k) is

                                 2 (k - 1) k n!
                       ----------------------------------
                                             2
                       n (n - 1) (n - 2) (k!)  (n - 2 k)!

       Check! (or believe me). This concludes the proof of Amitai Regev's

                                conjecture. QED.

                      this took, 0.094, second of CPU time