For a representation of a compact Lie group, the $n$th moment of the trace of that representation against the Haar measure is the dimension of the invariant subspace of the $n$th tensor power. The sequence of moments determines the distribution of the trace.

For the adjoint representations of $G_2,F_4$, and $E_7$, the $0$th through $7$th moments are all 1,0,1,1,5,16,80,436.

Is there a conceptual reason for this, or is it just a numerical coincidence?

Nick Katz asked this question in his graduate class.

One possible way to make this more conceptual is to note that there is more structure on the vector spaces of invariants than just their dimensions. There is an action of $S_n$, and maps front the $n$th invariants tensor the $m$th invariants to the $n+m$th invariants. Is this structure the same for these three groups? Is there a single structure with a conceptual definition that contains all three?

Some potentially related sequences, where the first $6$ elements agree with this one, are the moment sequences of the adjoint representations of $SP_6$ and $E_8$, and the sequence with exponential generating function $e^{-\int_{0}^x \log(1-y)dy}$

Is there some kind of stabilization phenomenon occurring where high-dimensional exceptional groups, if they existed, would also agree with this moment sequence? If so it doesn't seem to agree with the first few stable moments of the adjoint representation of any sequence of classical groups. For $A_n$ the moment sequence is $1,0,1,2,9, 44, 265, 1854$ (counting derangments) and $C_n$ and possibly the others have moment sequence $1,0,1,1,6,22,130, 822$ (counting graphs where each vertex has two edges, with multiple edges but without loops).