Let $G$ be reductive group over a field of characteristic $0$ ($GL_n$ fine for this question). Let $V$ be a linear representation of $G$. Then $G$ acts on the tensor algebra $T(V) = \bigoplus_{n \ge 0} V^{\otimes n}$ which is graded by $\deg(V) = 1$. Now form the Hilbert series of the ring of invariants:

$H(t) = \sum_{n \ge 0} \dim (V^{\otimes n})^G t^n$.

I believe I have a proof that this is a D-finite function, i.e., the derivatives of $H(t)$ with respect to $t$ form a finite-dimensional vector space over the field of rational functions. Is this result already stated in the literature?