A347106 - OEIS

%I #38 Jan 31 2022 08:37:02

%S 1,0,0,2,9,24,160,1350,10353,89936,910656,10070730,120546745,

%T 1566125352,21934589664,329037515534,5264316535905,89493067364640,

%U 1610885172539008,30606819613112466,612136012448309481,12854856587833586360,282806860558105285920

%N Number of derangements of [n] having an even number of 2-cycles.

%H Alois P. Heinz, <a href="/A347106/b347106.txt">Table of n, a(n) for n = 0..450</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Derangement">Derangement</a>.

%F E.g.f.: (exp(-x)+exp(-x*(x+1)))/(2-2*x).

%F a(n) = A000166(n) - A248087(n).

%F a(n) = Sum_{k=0..floor(n/4)} A162974(n,2*k).

%F a(n) mod 2 = A121262(n).

%e a(3) = 2: (123), (132).

%e a(4) = 9: (12)(34), (13)(24), (14)(23), (1234), (1243), (1324), (1342), (1423), (1432).

%p b:= proc(n, t) option remember; `if`(n=0, t, add(b(n-j,

%p      `if`(j=2, 1-t, t))*binomial(n-1, j-1)*(j-1)!, j=2..n))

%p     end:

%p a:= n-> b(n, 1):

%p seq(a(n), n=0..27);

%Y Cf. A000166, A088336, A121262, A162974, A248087.

%K nonn

%O 0,4

%A _Alois P. Heinz_, Jan 27 2022