A161122 - OEIS

%I #14 Sep 08 2022 08:45:45

%S 0,0,2,18,180,2100,28350,436590,7567560,145945800,3101348250,

%T 72020198250,1814908995900,49332526343100,1438865351673750,

%U 44826189802143750,1485668004871050000,52196469237802890000,1937793920453432291250,75801938653031321981250,3116301922402398792562500

%N Number of cycles with entries of the same parity in all fixed-point-free involutions of {1,2,...,2n}.

%H Vincenzo Librandi, <a href="/A161122/b161122.txt">Table of n, a(n) for n = 0..300</a>

%F a(n) = n(n-1)(2n-3)!!.

%F a(n) = Sum_{k>=0} k*A161121(n,k).

%F D-finite with recurrence (-n+2)*a(n) +n*(2*n-3)*a(n-1)=0. - _R. J. Mathar_, Jul 26 2022

%e a(2)=2 because in the 3 permutations (12)(34), (13)(24), (14)(23) we have a total of 2 cycles with entries of the same parity.

%p seq(n*(n-1)*(product(2*j-1, j = 1 .. n-1)), n = 0 .. 18);

%t Table[n (n - 1) (2 n -3)!!, {n, 0, 20}] (* _Vincenzo Librandi_, Jul 21 2017 *)

%o (Magma) DoubleFactorial:=func< n | &*[n..2 by -2] >; [ n*(n-1)*DoubleFactorial(2*n-3): n in [0..22]]; // _Vincenzo Librandi_, Jul 21 2017

%Y Cf. A161119, A161120, A161121.

%K nonn

%O 0,3

%A _Emeric Deutsch_, Jun 02 2009