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alternating factorials
The alternating factorial  $a(n)$  of an integer  $n>0$, is equal to
\[n!-(n-1)!+(n-2)!-(n-3)!+\cdots\pm1!,\]
where the sign of the last term depends on the parity of  $n$. By definition,  $a(0)=1$.

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.

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here
remainders of alternating factorials
Useful links Mathworld, Alternating Factorial
Wikipedia, Alternating factorial
OEIS, Sequence A005165
Alternating factorials can also be... (you may click on names or numbers)

a-pointer 101 aban 19 101 619 alternating 101 amenable 101 4421 326981 36614981 apocalyptic 4421 arithmetic 19 101 619 4421 35899 326981 3301819 c.decagonal 101 c.triangular 19 Chen 19 101 4421 congruent 101 4421 326981 Cunningham 101 Curzon 326981 36614981 cyclic 19 101 619 4421 35899 326981 3301819 deficient 19 101 619 4421 35899 326981 3301819 dig.balanced 19 35899 36614981 Duffinian 326981 economical 19 101 619 4421 35899 326981 3301819 emirp 3301819 emirpimes 36614981 equidigital 19 101 619 4421 35899 326981 3301819 esthetic 101 evil 101 619 35899 3301819 442386619 fibodiv 19 good prime 101 happy 19 Harshad 5784634181 hex 19 iban 101 4421 inconsummate 326981 junction 101 3301819 lucky 619 magnanimous 101 metadrome 19 modest 19 nialpdrome 4421 oban 19 619 odious 19 4421 326981 36614981 palindromic 101 palprime 101 pandigital 19 partition 101 pernicious 19 4421 326981 Pierpont 19 plaindrome 19 35899 prime 19 101 619 4421 35899 3301819 repfigit 19 semiprime 326981 36614981 sliding 101 strobogrammatic 101 619 strong prime 101 4421 super-d 19 619 326981 3301819 twin 19 101 619 4421 35899 uban 19 undulating 101 upside-down 19 weak prime 19 619 35899 3301819