On the Generalization of Factoriangular Numbers
Romer C. Castillo *
Batangas State University, Batangas City, Philippines.
*Author to whom correspondence should be addressed.
Abstract
A factoriangular number is a sum of a factorial and its corresponding triangular number. This paper presents some forms of the generalization of factoriangular numbers. One generalization is the \(n^{(m)}\) -factoriangular number which is of the form \((n!)^{m}\) + \(S_m(n)\), where \((n!)^{m}\) is the \(m\)th power of the factorial of \(n\) and \(S_m(n)\) is the sum of the \(m\)-powers of \(n\). This generalized form is explored for the different values of the natural number \(m\). The investigation results to some interesting proofs of theorems related thereto. Two important formulas were generated for \((n)^{m}\) -factoriangular number: \(Ft_{n^{(m)}}\) = \(Ft_{n^{(2k)}}\) = \((n!)^{2k}\) + \(2n+1\over2k+1\)\([n^{2k-2}+P(n^{2k-3})]T_n\) for even \(m=2k\), and \(Ft_{n^{(m)}}\) = \(Ft_{n^{(2k+1)}}\) = \((n!)^{2k+1}\) + \(n(n+1)\over k+1\)\([n^{2k-2}+P(n^{2k-3})]T_n\) for odd \(m=2k+1\)
Keywords: Factoriangular numbers, generalized factoriangular numbers, factorial, triangular numbers, sums of powers, Faulhaber’s sums, integer sequences, number theory