A056311 Number of reversible strings with n beads using exactly four different colors.

0, 0, 0, 12, 120, 780, 4212, 20424, 93360, 409380, 1749780, 7338792, 30394560, 124705140, 508291812, 2061607224, 8332140720, 33585777060, 135116412660, 542785800072, 2178110589600, 8733345234900

Offset: 1

Author

Marks R. Nester

Keywords

• nonn

Comments

A string and its reverse are considered to be equivalent.

Examples

			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
		

References

• 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]

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A032121.

Column 4 of A305621.

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

Programs

Mathematica

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

Formula

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

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]

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