%I #7 Jan 04 2017 14:31:55
%S 1,2,2,6,22,50,274,598,4486,9570,90914,191398,2201078,4593554,
%T 62012978,128619510,1993602406,4115824322,72026925634,148169675590,
%U 2889308674006
%N Number of symmetric reduced rearrangement maps.
%C a(n) is the number of reduced rearrangement maps on n blocks. A rearrangement map is a signed permutation, e.g., +2 -1 -3. If the permutation contains (i)(i+1) or -(i+1)-(i) for any i, then it is not reduced. The map a is symmetric if a=a^(AI) and a^A = a^I where A and I are the rotation involutions.
%D J. Burns, Counting a Class of Signed Permutations and Chord Diagrams related to DNA Rearrangement, Preprint.
%H J. Burns, <a href="http://jtburns.myweb.usf.edu/tables/rearrangement_maps.html">Table of Rearrangement Maps and Patterns for n = 1, 2, and 3</a>.
%F a(n) = round( 2^n * e^(-1/4) * ( 1 - (1 + (-1)^n)/(4n) ) * floor(n/2)! )
%F a(2k+1) = 2*a(2k) + a(2k-1) and a(2k) = (2k-1)*a(2k-1)+(2k-2)*a(2k-3)
%F a(n) ~ e^(-1/4) * 2^n * floor(n/2)!.
%F Conjecture: (-2*n+9)*a(n) -4*a(n-1) +(2*n-3)*(2*n-7)*a(n-2) -4*a(n-3) +2*(2*n-5)*(n-4)*a(n-4)=0. - _R. J. Mathar_, Jan 04 2017
%e For n=0 the a(0)=1 solution is { ∅ }
%e For n=1 the a(1)=2 solutions are { +1, -1 }
%e For n=2 the a(2)=2 solutions are { +2+1, -1-2 }
%e For n=3 the a(3)=6 solutions are { +3-2+1, -1+2-3, +3+2+1, -1-2-3, +1-2+3, -3+2-1 }
%t Table[Round[2^n*Exp[-1/4]*(1-(1+(-1)^n)/(4 n))*Floor[n/2]!],{n,1,20}]
%Y A271217 / A271216 ~ e^(-1/4).
%Y Cf. A271212, A271214
%K nonn,easy
%O 0,2
%A _Jonathan Burns_, Apr 13 2016