The alternating factorial is defined as the sum of consecutive factorials with alternating signs, a(n) congruent sum_(k = 1)^n (-1)^(n - k) k!. They can be given in closed form as a(n) = (-1)^n[-1 - e Ei(-1) + (-1)^n E_(n + 2)(1) Γ(n + 2)], where Ei(x) is the exponential integral, E_n(x) is the E_n-function, and Γ(x) is the gamma function. The alternating factorial will is implemented in the Wolfram Language as AlternatingFactorial[n].
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