How to generate a nonsquare Vandermonde matrix?

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Enter matrix.vandermonde?? to see the source code (3 lines long). We can extend the functionality:

def my_vandermonde(v, ncols=None, ring=None):
    def entries(i, j):
        return v[i]**j
    if ncols is None:
        ncols = len(v)
    return matrix(entries, nrows=len(v), ncols=ncols, ring=ring)

The relative inefficiency here (taking powers instead of repeatedly multiplying) is copied from the original.

Then you can do for instance:

sage: my_vandermonde([2,3,4], 4)
[ 1  2  4  8]
[ 1  3  9 27]
[ 1  4 16 64]