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How to Produce Lagrange Inversion Coefficients?

I am new to the Mathematica I want to Produce the Lagrange Inversion Coefficients by using Bell Polynomials.enter image description here In the following image you can see 7 Coefficients which is calculated. You can also see the An formula for calculating these coefficients by using Bell Polynomials. Bn-1,k is the Bell Polynomial. I want to produce the coefficients more than A7 I don't know how many times I should go further it depends on how much it converges to my result. I believe it won't be hard to producing them but I have few knowledge in Mathematica yet.