This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A161120 #11 Jul 26 2022 13:46:49
%S A161120 0,1,4,27,240,2625,34020,509355,8648640,164189025,3445942500,
%T A161120 79222218075,1979900722800,53443570205025,1549547301802500,
%U A161120 48028060502296875,1584712538529120000,55458748565165570625
%N A161120 Number of cycles with entries of opposite parities in all fixed-point-free involutions of {1,2,...,2n}.
%F A161120 a(n)=n^2*(2n-3)!!
%F A161120 a(n)=Sum(k*A161119(n,k), k=0..n)
%F A161120 E.g.f.: x(1-x)/(1-2x)^(3/2). [From _Paul Barry_, Sep 13 2010]
%F A161120 E.g.f.: x/2*U(0) where U(k)= 1 + (2*k+1)/(1 - x/(x + (k+1)/U(k+1))) ; (continued fraction, 3-step). - _Sergei N. Gladkovskii_, Sep 25 2012
%F A161120 D-finite with recurrence a(n) +(-2*n-1)*a(n-1) +a(n-2) +(-2*n+7)*a(n-3)=0. - _R. J. Mathar_, Jul 26 2022
%e A161120 a(2)=4 because in the 3 fixed-point-free involutions of {1,2,3,4}, namely (12)(34), (13)(24), (14)(23), we have a total of 4 cycles with entries of opposite parities.
%p A161120 seq(n^2*(product(2*j-1, j = 1 .. n-1)), n = 0 .. 18);
%Y A161120 Cf. A161119, A161121, A161122.
%K A161120 nonn
%O A161120 0,3
%A A161120 _Emeric Deutsch_, Jun 02 2009