In mathematics, an alternating factorial is the absolute value of the alternating sum of the first n factorials.

This is the same as their sum, with the odd-indexed factorials multiplied by −1 if n is even, and the even-indexed factorials multiplied by −1 if n is odd, resulting in an alternation of signs of the summands (or alternation of addition and subtraction operators, if preferred). To put it algebraically,

$\backslash mathrm\{af\}(n)\; =\; \backslash sum\_\{i\; =\; 1\}^n\; (-1)^\{n\; -\; i\}i!$

or with the recurrence relation

$\backslash mathrm\{af\}(n)\; =\; n!\; -\; \backslash mathrm\{af\}(n\; -\; 1)$

in which af(1) = 1.

The first few alternating factorials are

1, 1, 5, 19, 101, 619, 4421, 35899, 326981, 3301819, 36614981, 442386619, 5784634181, 81393657019

For example, the third alternating factorial is 1! + −(2!) + 3! = 5, or if preferred, 1! − 2! + 3! The fourth alternating factorial is −(1!) + 2! + −(3!) + 4! = 19. Regardless of the parity of n, the summand n − 1 is given a negative sign and the signs of the lower-indexed summands are alternated accordingly.

This pattern of alternation ensures the resulting sums are all positive integers. Changing the rule so that either the odd- or even-indexed summands are given negative signs (regardless of the parity of n) changes the signs of the resulting sums but not their absolute values.

Except for n = 1, the factorial of n and the alternating factorial of n are coprime. Miodrag Zivković proved in 1999 that there are only a finite number of alternating factorials that are also prime numbers. The largest alternating factorial known to be a prime is the alternating factorial of 661, approximately 7.818097272875 × 10¹⁵⁷⁸.

Reference

• Yves Gallot, Is the number of primes $\{1\; \backslash over\; 2\}\backslash sum\_\{i\; =\; 0\}^\{n\; -\; 1\}\; i!$ finite?

Factorial and binomial topics