A056312 Number of reversible strings with n beads using exactly five different colors.

0, 0, 0, 0, 60, 900, 8400, 63000, 417120, 2551560, 14804700, 82764900, 450518460, 2404510500, 12646078200, 65771496000, 339165516120, 1737486149760, 8855359634100, 44952367981500, 227475768907860, 1148269329527100, 5785013373810000, 29100047092479000

Offset: 1

Author

Marks R. Nester

Keywords

• nonn
• easy

Comments

A string and its reverse are considered to be equivalent.

Examples

			For n=5, the 60 rows are 60 permutations of ABCDE that do not include any mutual reversals.  Each of the 60 chiral pairs, such as ABCDE-EDCBA, is then counted just once.
		

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
• Index entries for linear recurrences with constant coefficients, signature (13,-45,-75,695,-575,-3195,5595,4706,-14918,2160,12840,-7200).

Crossrefs

Cf. A005418, A032120, A032121, A032122.

Column 5 of A305621.

Equals (A001118 + A056456) / 2 = A001118 - A305625 = A305625 + A056456.

Programs

Magma

[60*(StirlingSecond(n, 5)+StirlingSecond(Ceiling(n/2), 5)): n in [1..30]]; // Vincenzo Librandi, Sep 30 2018

Mathematica

k=5; Table[(StirlingS2[i,k]+StirlingS2[Ceiling[i/2],k])k!/2,{i,30}] (* Robert A. Russell, Nov 25 2017 *) adapted
CoefficientList[Series[-60*x^4*(120*x^7 - 17*x^6 - 50*x^5 - 32*x^4 + 20*x^3 + 10*x^2 - 2*x - 1)/((x - 1)*(2*x - 1)*(2*x + 1)*(3*x - 1)*(4*x - 1)*(5*x - 1)*(2*x^2 - 1)*(3*x^2 - 1)*(5*x^2 - 1)), {x, 0, 30}], x] (* Stefano Spezia, Sep 29 2018 *)

PARI

a(n) = 60*(stirling(n, 5, 2) + stirling(ceil(n/2), 5, 2)); \\ Altug Alkan, Sep 27 2018

Formula

a(n) = A032122(n) - 5*A032121(n) + 10*A032120(n) - 10*A005418(n+1) + 5.

G.f.: -60*x^5*(120*x^7 - 17*x^6 - 50*x^5 - 32*x^4 + 20*x^3 + 10*x^2 - 2*x - 1)/((x - 1)*(2*x - 1)*(2*x + 1)*(3*x - 1)*(4*x - 1)*(5*x - 1)*(2*x^2 - 1)*(3*x^2 - 1)*(5*x^2 - 1)). [Colin Barker, Sep 03 2012]

a(n) = k! (S2(n,k) + S2(ceiling(n/2),k)) / 2, where k=5 is the number of colors and S2 is the Stirling subset number. - Robert A. Russell, Sep 25 2018