calculus and analysis - Derivative of generating function (Example from documentation)

Bug introduced in 9.0 or earlier and persisting through 11.0.1 or later

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In the documentation for GeneratingFunction, the following example is given under Examples -> Properties & Relations -> Derivative:

GeneratingFunction[n f[n], n, z]
(*z (GeneratingFunction^(0,0,1))[f[n],n,z]*)

When I evaluate the input, however, I obtain something different, namely:

GeneratingFunction[n f[n], n, z]
(*z GeneratingFunction[(1 + n) f[1 + n], n, z]*)

For me the first output would be preferable; indeed I would like Mathematica to express any expression of the form

GeneratingFunction[p[n] f[n], n, z] (*p[n] an explicit polynomial in n*)

in terms of derivatives of the generating function of the sequence f[n]. I think this is the intended behaviour of GeneratingFunction and certainly what I remember from v8.

Question: Is there a way to achieve this behaviour?

I am currently using version 9.0 on a 64bit Linux computer.