For example, a(5) = 5! - 4! + 3! - 2! + 1! and a(4) = 4! - 3! + 2! - 1!.
The first alternating factorials are 1, 5, 19, 101, 619, 4421, 35899, 326981, 3301819, 36614981, 442386619, 5784634181, 81393657019, 1226280710981, 19696509177019, 335990918918981.