I am new to Mathematica. I want to produce the Lagrange inversion coefficients by using Bell polynomials.

In the following image you can see expressions for the first seven coefficients $ A_1$ to $ A_7$ using Bell Polynomials. There is also a general expression in term of Bell polynomials $ B_{n-1,\,k}$

I want to produce coefficients higher than $ A_7$ . I don’t know how many times I should go further; it will depend on how quickly I get convergence to my result.

I believe it shouldn’t be hard to produce coefficients higher than $ A_7$ , but I have don’t have enough knowledge of Mathematica yet to do myself.