Mathematical models for vacuum tubes?
The general small-signal model of a tube is pretty much outlined at the link Madmanguruman posted: a current source in parallel with a plate resistance and the associated inter-electrode capacitances.  

For non-linear time domain analysis, the situation is more complicated as outlined here.  As the article states, it's possible to derive a mathematical model of a triode or pentode based on the Langmuir-Child law, but that model doesn't accurately represent the real behavior of a tube in certain areas of its operation.  The best models are "phenomenological" i.e. designed to fit a tube's actual performance curves as closely as possible without regard to underlying physics.  

There is a program I've used available that will let you take published tube curves, fit the model to the curves, and then spit out a SPICE subcircuit.  It works well for triodes, I don't know if it can be used for pentodes though.  There are also many, many ready-to-run SPICE models of varying quality for a variety of different tubes on the web.