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[binomial,block,delta,determinant,do,expand,genmatrix,kron_delta,return]

Run Example
(%i1)diffetadet(n,eta):=block(    [h,i,j,m],        h[i,j]:=kron_delta(i,j+1),    m: genmatrix(h,n+1,n+1),    for i:1 thru n+1 do (        m[1,i]: binomial(i,n-1)*eta(k+i)    ),    for i:2 thru n+1 do (        for j:i thru n+1 do (            m[i,j]: binomial(n,j)*eta(j-i+1)        )    ),    return(m),        return((-1)^n*expand(determinant(m))));
(%o1) diffetadet(n, eta) := block([h, i, j, m], h     := kron_delta(i, j + 1), 
                                                 i, j
m : genmatrix(h, n + 1, n + 1), for i thru n + 1 
do m     : binomial(i, n - 1) eta(k + i), 
    1, i
for i from 2 thru n + 1 do (for j from i thru n + 1 
do m     : binomial(n, j) eta(j - i + 1)), return(m), 
    i, j
            n
return((- 1)  expand(determinant(m))))
(%i2) diffetadet(2,eta);
                  [ eta(k + 1)  2 eta(k + 2)  3 eta(k + 3) ]
                  [                                        ]
(%o2)             [     1          eta(1)          0       ]
                  [                                        ]
                  [     0            1             0       ]
(%i3) 
Run Example
diffetadet(n,eta):=block(    [h,i,j,m],        h[i,j]:=kron_delta(i,j+1),    m: genmatrix(h,n+1,n+1),    for i:1 thru n+1 do (        m[1,i]: binomial(i,n-1)*eta(k+i)    ),    for i:2 thru n+1 do (        for j:i thru n+1 do (            h[i,j]: eta(j-i+1)        )    ),    return(m),        return((-1)^n*expand(determinant(m))));
(%o1) diffetadet(n, eta) := block([h, i, j, m], h     := kron_delta(i, j + 1), 
                                                 i, j
m : genmatrix(h, n + 1, n + 1), for i thru n + 1 
do m     : binomial(i, n - 1) eta(k + i), 
    1, i
for i from 2 thru n + 1 do (for j from i thru n + 1 
                                                   n
do h     : eta(j - i + 1)), return(m), return((- 1)  expand(determinant(m))))
    i, j
(%i2) diffetadet(1,eta);
                          [ eta(k + 1)  eta(k + 2) ]
(%o2)                     [                        ]
                          [     1           0      ]
(%i3) 
Run Example
diffetadet(n,eta):=block(    [h,i,j,m],        h[i,j]:=kron_delta(i,j+1),    m: genmatrix(h,n+1,n+1),    for i:1 thru n+1 do (        m[1,i]: binomial(n,i-1)*eta(k+i)    ),    for i:2 thru n+1 do (        for j:i thru n+1 do (            m[i,j]: binomial(n-i+2,j-i+1)*eta(j-i+1)        )    ),    return((-1)^n*expand(determinant(m))));
(%o1) diffetadet(n, eta) := block([h, i, j, m], h     := kron_delta(i, j + 1), 
                                                 i, j
m : genmatrix(h, n + 1, n + 1), for i thru n + 1 
do m     : binomial(n, i - 1) eta(k + i), 
    1, i
for i from 2 thru n + 1 do (for j from i thru n + 1 
do m     : binomial(n - i + 2, j - i + 1) eta(j - i + 1)), 
    i, j
            n
return((- 1)  expand(determinant(m))))
(%i2) expand(diffetadet(3,eta));
(%o2) eta(k + 4) - 3 eta(1) eta(k + 3) - 3 eta(2) eta(k + 2)
        2
 + 6 eta (1) eta(k + 2) - eta(3) eta(k + 1) + 6 eta(1) eta(2) eta(k + 1)
        3
 - 6 eta (1) eta(k + 1)
(%i3) 
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