cp's OEIS Frontend

A347106 Number of derangements of [n] having an even number of 2-cycles.

%I A347106 #38 Jan 31 2022 08:37:02
%S A347106 1,0,0,2,9,24,160,1350,10353,89936,910656,10070730,120546745,
%T A347106 1566125352,21934589664,329037515534,5264316535905,89493067364640,
%U A347106 1610885172539008,30606819613112466,612136012448309481,12854856587833586360,282806860558105285920
%N A347106 Number of derangements of [n] having an even number of 2-cycles.
%H A347106 Alois P. Heinz, <a href="/A347106/b347106.txt">Table of n, a(n) for n = 0..450</a>
%H A347106 Wikipedia, <a href="http://en.wikipedia.org/wiki/Derangement">Derangement</a>.
%F A347106 E.g.f.: (exp(-x)+exp(-x*(x+1)))/(2-2*x).
%F A347106 a(n) = A000166(n) - A248087(n).
%F A347106 a(n) = Sum_{k=0..floor(n/4)} A162974(n,2*k).
%F A347106 a(n) mod 2 = A121262(n).
%e A347106 a(3) = 2: (123), (132).
%e A347106 a(4) = 9: (12)(34), (13)(24), (14)(23), (1234), (1243), (1324), (1342), (1423), (1432).
%p A347106 b:= proc(n, t) option remember; `if`(n=0, t, add(b(n-j,
%p A347106      `if`(j=2, 1-t, t))*binomial(n-1, j-1)*(j-1)!, j=2..n))
%p A347106     end:
%p A347106 a:= n-> b(n, 1):
%p A347106 seq(a(n), n=0..27);
%Y A347106 Cf. A000166, A088336, A121262, A162974, A248087.
%K A347106 nonn
%O A347106 0,4
%A A347106 _Alois P. Heinz_, Jan 27 2022