We use a modulation-doped double barrier heterostructure to fabricate a resonant tunneling single electron transistor. Irregular Coulomb blockade oscillations are observed when the gate voltage is swept to vary one-by-one the number of electrons in the dot close to 'pinch-off'. The oscillation period is not regular, and generally becomes longer as the electron number is decreased down to zero, reflecting the growing importance of electron-electron interactions and size quantization. Negative differential resistance associated with resonant tunneling through zero-dimensional states is pronounced for a dot holding just a few electrons. The temperature dependence of the Coulomb blockade oscillations and that for the negative differential resistance are not the same. This highlights the different effects of charging and resonant tunneling on the transport characteristics.