This text deals with combinatorics, finite calculus, formal power series and generating functions, recurrence relations, symbolic calculus and approximation formulae for finite sums. There are several books on these arguments, even though few are covering all of them. This books wants to be an introductory text which leads the reader from basic knowledge to deeper insights into specific topics, while maintaining an accurate mathematical style. The aim here is to provide a strategy in order to classify and solve problems, avoiding tricks and ad hoc solutions, thanks to a rigorous and at the...