%I #26 Oct 10 2018 09:32:03

%S 0,0,0,12,120,780,4212,20424,93360,409380,1749780,7338792,30394560,

%T 124705140,508291812,2061607224,8332140720,33585777060,135116412660,

%U 542785800072,2178110589600,8733345234900

%N Number of reversible strings with n beads using exactly four different colors.

%C A string and its reverse are considered to be equivalent.

%D M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

%H Vincenzo Librandi, <a href="/A056311/b056311.txt">Table of n, a(n) for n = 1..1000</a>

%F Equals A032121(n) - 4*A032120(n) + 6*A005418(n+1) - 4.

%F G.f.: 12*x^4*(3*x+1)*(8*x^4-3*x^3-2*x^2-x+1)/ ((x-1) * (4*x-1) * (3*x-1) * (2*x+1) * (2*x -1) * (3*x^2-1) * (2*x^2-1)). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009 [Corrected by _R. J. Mathar_, Sep 16 2009]

%F a(n) = k! (S2(n,k) + S2(ceiling(n/2),k)) / 2, where k=4 is the number of colors and S2 is the Stirling subset number. - _Robert A. Russell_, Sep 25 2018

%e For n=4, the 12 rows are 12 permutations of ABCD that do not include any mutual reversals. Each of the 12 chiral pairs, such as ABCD-DCBA, is then counted just once. - _Robert A. Russell_, Sep 25 2018

%t k=4; Table[(StirlingS2[i,k]+StirlingS2[Ceiling[i/2],k])k!/2,{i,k,30}] (* _Robert A. Russell_, Nov 25 2017 *)

%t CoefficientList[Series[12 x^3 (3 x + 1) (8 x^4 - 3 x^3 - 2 x^2 - x + 1) / ((x - 1) (4 x - 1) (3 x - 1) (2 x + 1) (2 x - 1) (3 x^2 - 1) (2 x^2 - 1)), {x, 0, 33}], x] (* _Vincenzo Librandi_, Sep 26 2018 *)

%Y Cf. A032121.

%Y Column 4 of A305621.

%Y Equals (A000919 + A056455) / 2 = A000919 - A305624 = A305624 + A056455.

%K nonn

%O 1,4

%A _Marks R. Nester_