  We present a universal formula for the dimension of the Cartan powers of the
adjoint representation of a complex simple Lie algebra (i.e., a universal
formula for the Hilbert functions of homogeneous complex contact manifolds), as
well as several other universal formulas. These formulas generalize formulas of
Vogel and Deligne and are given in terms of rational functions where both the
numerator and denominator decompose into products of linear factors with
integer coefficients. We also discuss some consequences of the formulas
including a relation with Scorza varieties.
