Remarks on Bell and higher order Bell polynomials and numbers

We recover a recurrence relation for representing in an easy form the coefficients $ A_{n,k} $ of the Bell polynomials, which are known in literature as the partial Bell polynomials. Several applications in the framework of classical calculus are derived, avoiding the use of operational techniques. Furthermore, we generalize this result to the coefficients $ A^{[2]}_{n,k} $ of the second-order Bell polynomials, i.e. of the Bell polynomials relevant to nth derivative of a composite function of the type f(g(h(t))). The second-order Bell polynomials $ B_n^{[2]} $ and the relevant Bell numbers $ b_n^{[2]} $ are introduced. Further extension of the nth derivative of M-nested functions is also touched on