binomial determinant

[benoit]

Hi,

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.


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pari-gp check :

?  z(n)=prime(n)
? a(n)=matdet(matrix(n,n,i,j,binomial(z(j)*i,j)))
? for(n=1,10,print1(a(n)-prod(k=1,n,z(k)^k),","))
0,0,0,0,0,0,0,0,0,0,
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Benoit Cloitre