n | 1 | 2 | 3 | 4 | 5 subfactorial(n) | 0 | 1 | 2 | 9 | 44

Γ(n + 1, -1)/e

(-1)^n (1/n - 2/n^2 + 5/n^3 - 15/n^4 + 52/n^5 + O((1/n)^6)) + e^(n (log(n) - 1) - 1) (sqrt(2 π) sqrt(n) + 1/6 sqrt(π/2) sqrt(1/n) + 1/144 sqrt(π/2) (1/n)^(3/2) - (139 sqrt(π/2) (1/n)^(5/2))/25920 - (571 sqrt(π/2) (1/n)^(7/2))/1244160 + (163879 sqrt(π/2) (1/n)^(9/2))/104509440 + O((1/n)^(11/2)))

d/dn(subfactorial(n)) = (G_(2, 3)^(3, 0) (-1|1, 1 0, 0, n + 1))/e + i π subfactorial(n)

subfactorial(n) = (E_(-n)(-1) (-1)^(-1 + n))/e

subfactorial(n) = (Q(1 + n, -1) Γ(1 + n))/e

subfactorial(n) = (Γ(n + 1, -1, 0) + Γ(n + 1))/e for (n not element Z or n>=0)

subfactorial(n) = ((Q(n + 1, -1, 0) + 1) Γ(n + 1))/e for (n not element Z or n>=0)

subfactorial(n)∝(-1)^n - (-1)^n/e + (n Γ(n))/e - ((-1)^n sum_(k=0)^∞ sum_(j=1)^∞ ((-j)^k n^(-k))/(j!))/e for abs(n)->∞

subfactorial(n)∝(n n! + (-1)^n sum_(k=0)^∞ (-1)^k n^(-k) + (-1)^n sum_(k=0)^∞ sum_(j=0)^∞ ((-1)^k (2 + j)^k n^(-k))/((1 + j)!))/(e n) for abs(n)->∞

subfactorial(n)∝2^(-floor((π + arg(n))/(2 π))) e^(-1 - n - i π floor((π + arg(n))/(2 π))) (-(-1)^n 2^floor((π + arg(n))/(2 π)) e^(n + i π floor((π + arg(n))/(2 π))) + (-1)^n 2^floor((π + arg(n))/(2 π)) e^(1 + n + i π floor((π + arg(n))/(2 π))) + exp(1/2 sum_(k=0)^∞ (n^(-1 - 2 k) B_(2 + 2 k))/(1 + 3 k + 2 k^2)) (e^(i π floor((π + arg(n))/(2 π))) n)^(1/2 + n) sqrt(2 π) csc^floor((π + arg(n))/(2 π))(n π) - (-1)^n 2^floor((π + arg(n))/(2 π)) e^(n + i π floor((π + arg(n))/(2 π))) sum_(k=0)^∞ sum_(j=1)^∞ ((-j)^k n^(-k))/(j!)) for ((n not element Z or n>=1) and abs(n)->∞)

subfactorial(n) = integral_0^∞ e^(-t) (-1 + t)^n dt for Re(n)>-1