The meaning of 3-j symbol is same as Clebsch-Gordan coefficient. So, we skip in here.

I am not going to construct the 6-j symbol from 3-j symbol. In here, I just state the meaning and usage in Mathematica.

The 6-j symbol is the coupling between 3 angular momenta, .

There are 2 ways to couple these 3 angular momenta. First,

the other way is

The 6-j symbol is

We can see that there are 4 vector-sum must satisfy.

If we draw a line to connect these 4 vector-sum, we have:

In Mathematica, there is a build in function

The 9-j symbol is the coupling between 4 angular momenta, .

The 9-j symbol can be used in coupling 2 nucleons, .

The 9-j symbol is

We can see, each row and column must satisfy the vector-sum.

Unfortunately, there is no build in function in Mathematica. The formula for 9-j symbol is

Where sum all possible value, which can be calculate using the 6 couplings inside the 3 6-j symbols.To check your result, the coupling between and to from a state is