Dimension of Harmonic Polynomial space from Tensor definition

I would like to calculate the dimension of the Harmonic Polynomial space using the following definition:
Let $P_n(overrightarrow{x})$ be a polynomial of degree n of the variable $overrightarrow{x}=(x_1,x_2,x_3)$; it can be wrote as $P_n(overrightarrow{x})=sum_{i_1,…,i_n} T_{n;i_1,…,i_n}x^{i_1}…x^{i_n}$ where $T_{n;i_1,…,i_n}$ is a symmetric tensor. If $P_n(overrightarrow{x})$ is harmonic, $nabla^2P_n(overrightarrow{x})=0$ implies $Tr(T_{n;i_1,…,i_n})=0$.
Hence $T_{n;i_1,…,i_n}$ is a symmetric traceless tensor.
From this property of $T$, can I deduce the dimension of the space?
Thank you!