The term "left factorial" is sometimes used to refer to the subfactorial !n, the first few values for n = 1, 2, ... are 1, 3, 9, 33, 153, 873, 5913, ... (OEIS A007489). Unfortunately, the same term and notation are also applied to the factorial sum L!n | = | sum_(k = 0)^(n - 1) k! | = | (-1)^n(n!)!(-n - 1) - !(-1) | = | ((-1)^n Γ(n + 1) Γ(-n, -1) - Γ(0, -1))/e | = | (i π + Ei(1) + Γ(n + 1, -1) E_(n + 1)(-1))/e, where Γ(z) is a gamma function, Ei(x) is the exponential integral, and E_n(x) is the E_n-function.
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