If you have an exploding N-sided die that adds another die when it's at max, the average A is:

    A = (1+2+...N+A)/N

This is the same as:

    A = (N(N+1) + A)/2N

This, in turn, can be simplified to:

    2N*A = N(N+1) + A

And that in turn, can be simplified to what Pat Ludwig has so kindly tabulated.

A similar method can be used to calculate averages when the highest roll of a die is substituted by tolling 2 (or more) of the same dice-type.