n | 1 | 2 | 3 | 4 | 5 superfactorial(n) | 1 | 2 | 12 | 288 | 34560

1 + 1/2 n (-1 - 2 gamma + log(2 π)) + 1/2 n^2 (-1 - gamma + π^2/6 + 1/4 (-1 - 2 gamma + log(2 π))^2) + 1/6 n^3 (π^2/3 + 1/4 (6 + 6 gamma - π^2) (1 + 2 gamma - log(2 π)) - 1/8 (1 + 2 gamma - log(2 π))^3 + polygamma(2, 1)) + 1/24 n^4 (π^4/15 + 1/12 (-6 - 6 gamma + π^2)^2 - 1/4 (6 + 6 gamma - π^2) (-1 - 2 gamma + log(2 π))^2 + 1/16 (-1 - 2 gamma + log(2 π))^4 + 3 polygamma(2, 1) - 2/3 (1 + 2 gamma - log(2 π)) (π^2 + 3 polygamma(2, 1))) + O(n^5) (Taylor series)

d/dn(superfactorial(n)) = 1/2 superfactorial(n) (-2 n + 2 (n + 1) polygamma(0, n + 1) - 1 + log(2 π))