This notebook contains the formula referred to in the final
example of the paper for the (conjecturally minimal) coefficient
of the monomial given there.

B[u_, v_]  :=   Binomial[u, v]

A1  :=  
      Function[s, 
        Sum[ B[s, l]^3 (Sum[ 
                B[l, j]^2  ( B[l,  j + 1]   -   B[l, j]),  {j, 
                  0, l}] ) ( 
              Sum[ B[s - l , i]^2  ( 
                    B[s - l,  i + 1]   -   B[s - l,  i]),  {i, 0, 
                  s - l}]), {l, 0, s}]]

A2  :=  
      Function[s,  
        Sum[ B[s, l]^2 B[s, l  + 1] ( 
              Sum[ B[l, 
                    j] (B[l, j + 1] B[l + 1, j + 2]  +  
                      B[l, j + 1] B[l + 1,  j + 1]  -  
                      2 B[l, j] B[l + 1, j + 1]),  {j, 0, l}]) ( 
              Sum[ B[s - l, 
                    i] (B[s - l,  i]  B[s - l - 1, i - 1]  -  
                      B[s - l,  i + 1] B[s - l - 1, i]),  {i, 0, 
                  s - l}]), {l, 0, s - 1}]]

A3  :=  
      Function[s,  
        Sum[ B[s, l]^2 B[s, l + 2] 
( 
              Sum[ B[l, j] ( 
                    B[l, j + 1] B[l + 2,  j + 2] -   B[l, j] B[l + 2, 
                          j + 1]),  {j, 0, l}] ) ( 
              Sum[B[s - l,  i]  B[s - l - 2,  i - 1] ( 
                    B[s - l, i + 1]  -  B[s - l, i] ),  {i, 0, 
                  s - l}]), {l, 0, s - 2}]]

A1[4]

A2[4]

A3[4]

A4  :=  
      Function[s,  
        Sum[ B[s, l + 1]^2 B[s, l] 
( 
              Sum[B[l + 1, j] ( 
                    B[l, j + 1] B[l + 1, j + 1]  +    B[l, j] B[l + 1, 
                          j + 1]  -  
                      2  B[l + 1, j]  B[l, j - 1]),  {j, 0, 
                  l + 1}] ) 
(Sum[ 
                B[s - l - 1, i] B[s - l,  
                    i + 1] (B[s - l - 1, i]  -  
                      B[s - l - 1, i + 1]),  {i, 0, s - l - 1}]), {l, 0, 
            s - 1}]]

A4[4]

A5  :=  
      Function[s,  
        Sum[ B[s, l] B[s, l + 1] B[s, l + 2] 
( 
              Sum[ B[l + 1, j] ( 
                    B[l,  j + 1] B[l + 2,  j + 2]   -   B[l, j] B[
                          l + 2, j + 1] ),  {j, 0, 
                  l}])
( 
              Sum[ B[s - l - 1, i] B[s - l,  
                    i + 1] (B[s - l - 2,  i] -  
                      B[s - l - 2,  i - 1]),  {i, 0, s - l - 1}]), {l, 
            0, s - 2}]]

A5[4]

A6  :=  
      Function[ s,  
        Sum[ B[s, l + 2]^2  B[s, l] 
( 
              Sum[ B[l + 2, j] ( 
                    B[l, j - 2] B[l + 2, j - 1]  -  
                      B[l, j - 1] B[l + 2, j]),  {j, 1, 
                  l + 2}] ) 
( 
              Sum[ B[s - l - 2, i] B[s - l,  i + 1] ( 
                    B[s - l - 2, i - 1]  -  B[s - l - 2, i]),  {i, 0, 
                  s - l - 2}]  ), {l, 0, 
            s - 2}]]

A6[4]

A  :=  
      Function[s,  
        A1[s]  +  A2[s]  +  A3[s]  +  A4[s]  +  A5[s]  +  
          A6[s]]

A[4]

J  :=  Function[s,  Sign[A[s]]]

For[i  =  1,  
      i  ≤  400,  i++,   If[J[i] <  0,  
        Print[NOTE, i]]];  Print[Finished];