binomial determinant

[Edwin Clark]

> 
> let z be any function : N-->N, let M_n be the nxn matrix 
> M_(i,j)=binomial(i*z(j) , j),  then I observed :
> 
> det M_n = prod(k=1,n, z(k)^k)
> 
> I'm looking for  a proof or anything related... Thanks.
> 
> Benoit Cloitre
> 

Maple gives a proof for n from 1 to 5:  I replace your functional values
z(j) by inteterminates z_i (or z[i] in Maple notation). Also I convert the
binomial expressions to factorial expressions. Then it holds without
restriction on the variables: 

> with(LinearAlgebra):
> for n from 1 to 5 do
> M:=Matrix(n,n,(i,j)->convert(binomial(i*z[j],j),factorial)):
> simplify(Determinant(M));
> print(n,%);
> end do:

                               1, z[1]


                                        2
                            2, z[1] z[2]


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


                             4          2     3
                      4, z[4]  z[1] z[2]  z[3]


                          4     3     5     2
                   5, z[4]  z[3]  z[5]  z[2]  z[1]


With some effort there should be a proof of this. :-)

--Edwin