We revisit the theory of Sheffer sequences by means of the formalism introduced in Rota and Taylor (SIAM J Math Anal 25(2):694–711, 1994) and developed in Di Nardo and Senato (Umbral nature of the Poisson random variables. Algebraic combinatorics and computer science, pp 245–256, Springer Italia, Milan, 2001, European J Combin 27(3):394–413, 2006). The advantage of this approach is twofold. First, this new syntax allows us noteworthy computational simplification and conceptual clarification in several topics involving Sheffer sequences, most of the open questions proposed in Taylor (Comput Math Appl 41:1085–1098, 2001) finds answer. Second, most of the results presented can be easily implemented in a symbolic language. To get a general idea of the effectiveness of this symbolic approach, we provide a formula linking connection constants and Riordan arrays via generalized Bell polynomials, here defined. Moreover, this link allows us to smooth out many results involving Bell Polynomials and Lagrange inversion formula

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Counting pattern avoiding ballot paths begins with a careful analysis of the pattern. Not the length, but the characteristics of the pattern are responsible for the difficulties in finding explicit solutions. Certain features, like overlap and difference in number of → and ↑ steps determine the solution of the recursion formula. If the recursion can be solved by a polynomial sequence, we apply the Finite Operator Calculus to find an explicit form of the solution in terms of binomial coefficients.