%I
%S 1,2,6,12,27,61,138,309,694,1560,3506,7877,17699,39770,89363,200796,
%T 451184,1013802,2277993,5118603,11501396,25843403,58069600,130481206,
%U 293188608,658788823,1480285049
%N Number of permutations in S_{2n} avoiding 123 and 1432 whose matrices are 180degree symmetric.
%C Trivially, this also counts 180degree symmetric permutations avoiding 321 and 4123, 123 and 3214, or 321 and 2341. For the other 140 pairs of patterns in S_3 and S_4, the sequence of symmetric permutations avoiding those patterns is either finite (as in 123 and 4321, by ErdosSzekeres) or counted by an easilyrecognized sequence such as alternating Fibonacci numbers, Catalan numbers, squares plus one, or the naturals.
%H G. C. Greubel, <a href="/A166963/b166963.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,1,1,1).
%F a(n) = 2a(n1) + a(n3) + a(n4)  a(n5).
%F G.f.: (x^3 + 2x^2 + 1)/(x^5  x^4  x^3  2x + 1).
%e For n=2, the a(2) = 6 solutions are 2143, 2413, 3142, 3412, 4231, and 4321. The two other 180degree symmetric permutations in S_4 are 1234 and 1324, both of which contain the pattern 123.
%t LinearRecurrence[{2,0,1,1,1},{1, 2, 6, 12, 27}, 50] (* _G. C. Greubel_, May 29 2016 *)
%K nonn
%O 0,2
%A David Lonoff and Jonah Ostroff (jonah.ostroff(AT)gmail.com), Oct 25 2009
%E Fixed typos caused by nonASCII symbol Jonah Ostroff (jonah.ostroff(AT)gmail.com), Oct 25 2009
