For a given positive integer $n$, I need to learn the number of $n\times n$ matrices of nonnegative integers with the following restrictions:

- The sum of each row and column is equal to $3$.
- Two matrices are considered equal if one can be obtained by permuting rows and/or columns.

For example, for $n=2$ there are two different matrices as follows:

$$M_1=\begin{pmatrix} 1&2\\2&1 \end{pmatrix},\qquad M_2=\begin{pmatrix} 3&0\\0&3 \end{pmatrix}.$$

For $n=3$ there are five different matrices as follows:

$$\begin{pmatrix} 1&1&1\\1&1&1\\1&1&1 \end{pmatrix},\quad \begin{pmatrix} 0&1&2\\1&2&0\\2&0&1 \end{pmatrix}, \quad \begin{pmatrix} 0&1&2\\1&1&1\\2&1&0 \end{pmatrix}, \quad \begin{pmatrix} 1&2&0\\2&1&0\\0&0&3 \end{pmatrix}, \quad \begin{pmatrix} 3&0&0\\0&3&0\\0&0&3 \end{pmatrix}.$$