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A161122 Number of cycles with entries of the same parity in all fixed-point-free involutions of {1,2,...,2n}.

%I A161122 #14 Sep 08 2022 08:45:45
%S A161122 0,0,2,18,180,2100,28350,436590,7567560,145945800,3101348250,
%T A161122 72020198250,1814908995900,49332526343100,1438865351673750,
%U A161122 44826189802143750,1485668004871050000,52196469237802890000,1937793920453432291250,75801938653031321981250,3116301922402398792562500
%N A161122 Number of cycles with entries of the same parity in all fixed-point-free involutions of {1,2,...,2n}.
%H A161122 Vincenzo Librandi, <a href="/A161122/b161122.txt">Table of n, a(n) for n = 0..300</a>
%F A161122 a(n) = n(n-1)(2n-3)!!.
%F A161122 a(n) = Sum_{k>=0} k*A161121(n,k).
%F A161122 D-finite with recurrence (-n+2)*a(n) +n*(2*n-3)*a(n-1)=0. - _R. J. Mathar_, Jul 26 2022
%e A161122 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 A161122 seq(n*(n-1)*(product(2*j-1, j = 1 .. n-1)), n = 0 .. 18);
%t A161122 Table[n (n - 1) (2 n -3)!!, {n, 0, 20}] (* _Vincenzo Librandi_, Jul 21 2017 *)
%o A161122 (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 A161122 Cf. A161119, A161120, A161121.
%K A161122 nonn
%O A161122 0,3
%A A161122 _Emeric Deutsch_, Jun 02 2009