# 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!