This paper presents an extensive investigation into several interrelated topics in mathematical analysis and number theory. The author revisits and builds upon known results regarding the Maclaurin power series expansions for a variety of functions and their normalized remainders, explores connections among central factorial numbers, the Stirling numbers, and specific matrix inverses, and derives several closed-form formulas and inequalities. Additionally, this paper reveals new insights into the properties of these mathematical objects, including logarithmic convexity, explicit expressions for certain quantities, and identities involving the Bell polynomials of the second kind.