http://www.research.att.com/~njas/sequences/A...Subfactorial or rencontres numbers, or derangements: number of permutations of n elements with no fixed points.
1, 0, 1, 2, 9, 44, 265, 1854, 14833, 133496, 1334961, 14684570, 176214841, 2290792932, 32071101049, 481066515734, 7697064251745, 130850092279664, 2355301661033953, 44750731559645106, 895014631192902121, 18795307255050944540
Euler not only gives the first ten or so terms of the sequence, he also proves both recurrences a(n)=(n-1)(a(n-1)+a(n-2)) and a(n)=na(n-1)+(-1)^n