%I
%S 1,1,3,11,65,513,5363,68219,1016481,17243105,327431363,6874989963,
%T 158118876449,3952936627361,106729080101235,3095142009014843,
%U 95949394016339393,3166329948046914369,110821547820208233731,4100397266856761733515
%N a(n) = (A001147(n) + A047974(n))/2.
%C Also, number of distinct Tsuro tiles which are digonal in shape and have n points per side. Turning over is not allowed. See A132100 for definition and comments.
%C See the Burns et al. papers for another interpretation.
%C From _Ross Drewe_, Mar 16 2008: (Start)
%C This is also the number of arrangements of n pairs which are equivalent under the joint operation of sequence reversal and permutations of labels. Assume that the elements of n distinct pairs are labeled to show the pair of origin, e.g., [1 1], [2 2]. The number of distinguishable ways of arranging these elements falls as the conditions are made more general:
%C a(n) = A000680: element order is significant and the labels are distinguishable;
%C b(n) = A001147: element order is significant but labels are not distinguishable, i.e. all label permutations of a given sequence are equivalent;
%C c(n) = A132101: element order is weakened (reversal allowed) and all label permutations are equivalent;
%C d(n) = A047974: reversal allowed, all label permutations are equivalent and equivalence class maps to itself under joint operation.
%C Those classes that do not map to themselves form reciprocal pairs of classes under the joint operation and their number is r(n). Then c = b  r/2 = b  (b  d)/2 = (b+d)/2. A formula for r(n) is not available, but formulas are available for b(n) = A001147 and d(n) = A047974, allowing an explicit formula for this sequence.
%C c(n) is useful in extracting structure information without regard to pair ordering (see example). c(n) terms also appear in formulas related to binary operators, eg, the number of binary operators in a kvalued logic that are invertible in 1 operation.
%C a(n) = (b(n) + c(n))/2, where b(n) = (2n)!/(2^n * n!), c(n) = sum(k=0,1,...floor(n/2)) (n!/((n2*k)! * k!)
%C For 3 pairs, the arrangement A = [112323] is the same as B = [212133] under the permutation of the labels [123] > [312] plus reversal of the elements, or vice versa. The unique structure common to A and B is {1 intact pair + 2 interleaved pairs}, where the order is not significant (contrast A001147). (End)
%H Jonathan Burns, <a href="http://shell.cas.usf.edu/~saito/DNAweb/SimpleAssemblyTable.txt">Assembly Graph Words  Single Transverse Component (Counts)</a>.
%H Jonathan Burns, Egor Dolzhenko, Natasa Jonoska, Tilahun Muche, and Masahico Saito, <a href="http://jtburns.myweb.usf.edu/assembly/papers/Graphs_and_DNA_Recomb_2011.pdf">FourRegular Graphs with Rigid Vertices Associated to DNA Recombination</a>, May 23, 2011.
%t Table[((2n1)!!+I^(n)*HermiteH[n,I/2])/2,{n,0,10}] (* _Jonathan Burns_, Apr 05 2016 *)
%Y Cf. A000680, A001147, A047974, A007769, A054499.
%Y Cf. A132100, A132101, A132102, A132103, A132104, A132105.
%K nonn,nice
%O 0,3
%A _Keith F. Lynch_, Oct 31 2007
%E Entry revised by _N. J. A. Sloane_, Nov 04 2011
