Recurrence Relations and Generating Functions of the Sequence of Sums of Corresponding Factorials and Triangular Numbers

This study established some recurrence relations and exponential generating functions of the sequence of factoriangular numbers. A factoriangular number is defined as a sum of corresponding factorial and triangular number. The proofs utilize algebraic manipulations with some known results from calculus, particularly on power series and Maclaurin’s series. The recurrence relations were found by manipulating the formula defining a factoringular number while the ascertained exponential generating functions were in the closed form.

Keywords – closed formula, factorial, factoriangular number, exponential generating function, recurrence relation, sequence, triangular number