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It's time for another round of WTF, The On-Line Encyclopedia of Integer Sequences?

You know how the factorial function has the recurrence

(n+1)! = (n+1) (n!)

and the 'subfactorial' function ---- which counts the derangements of length n --- has the recurrence

!(n+1) = (n+1) (!n) + (-1)^(n+1)

right?

Well, what happens if I put some other root of unity in for that -1? Like, let me say

f(n+1) = (n+1) (f(n)) + (i)^(n+1)

and for the hell of it set f(1) = 1. Then what do I get?

Aaaand the imaginary part of f(n) is apparently A186359, the "number of permutations of {1,2,...,n} having no up-down cycles". Can't find anything interestingly combinatorial for its real part, but it's (up to a sign flip) the egf of cos(x)/(1+x). (A009102).

You know how the factorial function has the recurrence

(n+1)! = (n+1) (n!)

and the 'subfactorial' function ---- which counts the derangements of length n --- has the recurrence

!(n+1) = (n+1) (!n) + (-1)^(n+1)

right?

Well, what happens if I put some other root of unity in for that -1? Like, let me say

f(n+1) = (n+1) (f(n)) + (i)^(n+1)

and for the hell of it set f(1) = 1. Then what do I get?

n f(n) 1 1 2 1 3 3-i 4 13-4i 5 65-19i 6 389-114i 7 2723-799i 8 21785-6392i 9 196065-57527i

Aaaand the imaginary part of f(n) is apparently A186359, the "number of permutations of {1,2,...,n} having no up-down cycles". Can't find anything interestingly combinatorial for its real part, but it's (up to a sign flip) the egf of cos(x)/(1+x). (A009102).