What is $\dim \text{Harm}^k(\mathbb{R}^n, \text{SO}_n(\mathbb{R}))$?

Question

A polynomial $f \in \mathbb{C}[x_1, \ldots, x_n]$ is called $G$-harmonic if $u(f) = 0$ for all $m > 0$ and all $G$-invariant homogeneous differential operators $u$ with constant coefficients of degree $m$.

Let $\text{Harm}(\mathbb{R}^n, G)$ be the space of $G$-harmonic polynomials on $\mathbb{R}^n$, and $\text{Harm}^m(\mathbb{R}^n, G)$ the subspace of $G$-harmonic, homogeneous polynomials of degree $m$.

Question. What is $\dim \text{Harm}^k(\mathbb{R}^n, \text{SO}_n(\mathbb{R}))$?

Answer 1

This is given in (equation 52) http://www.sciencedirect.com/science/article/pii/S0377042709001411

$$\dim=\frac{(n+2k-2)(n+k-3)!}{k!(n-2)!}$$