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

1,4

Comments

A string and its reverse are considered to be equivalent.

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

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

Example

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

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 *)

Crossrefs

Cf. A032121.

Column 4 of A305621.

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

Sequence in context: A133386 A305624 A056320 * A009050 A067358 A268634

Adjacent sequences: A056308 A056309 A056310 * A056312 A056313 A056314

Keyword

nonn

Author

Marks R. Nester

Status

approved