A056329 Number of reversible string structures with n beads using exactly five different colors.

0, 0, 0, 0, 1, 9, 76, 542, 3523, 21393, 123680, 690550, 3756151, 20042589, 105394296, 548123982, 2826435403, 14479204833, 73794961960, 374603884910, 1895632969351, 9568915372269, 48208452866816

Offset

1,6

Comments

A string and its reverse are considered to be equivalent. Permuting the colors will not change the structure.

Number of set partitions for an unoriented row of n elements using exactly five different elements. An unoriented row is equivalent to its reverse. - Robert A. Russell, Oct 14 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

Table of n, a(n) for n=1..23.

Index entries for linear recurrences with constant coefficients, signature (13, -48, -36, 551, -683, -1542, 3546, 80, -4280, 2400).

Formula

a(n) = A056324(n) - A056323(n).

Empirical g.f.: -x^5*(70*x^5 - 102*x^4 + 22*x^3 + 7*x^2 - 4*x + 1) / ((x-1)*(2*x-1)*(2*x+1)*(3*x-1)*(4*x-1)*(5*x-1)*(2*x^2-1)*(5*x^2-1)). - Colin Barker, Nov 25 2012

From Robert A. Russell, Oct 14 2018: (Start)

a(n) = (S2(n,k) + A(n,k))/2, where k=5 is the number of colors (sets), S2 is the Stirling subset number A008277 and A(n,k) = [n>1] * (k*A(n-2,k) + A(n-2,k-1) + A(n-2,k-2)) + [n<2 & n==k & n>=0].

a(n) = (A000481(n) + A304975(n)) / 2 = A000481(n) - A320528(n) = A320528(n) + A304975(n). (End)

Example

For a(6)=9, the color patterns are ABCDEA, ABCDBA, ABCCDE, AABCDE, ABACDE, ABCADE, ABCDAE, ABBCDE, and ABCBDE. The first three are achiral. - Robert A. Russell, Oct 14 2018

Mathematica

k=5; Table[(StirlingS2[n, k] + If[EvenQ[n], 3StirlingS2[n/2+2, 5] - 11StirlingS2[n/2+1, 5] + 6StirlingS2[n/2, 5], StirlingS2[(n+5)/2, 5] - 3StirlingS2[(n+3)/2, 5]])/2, {n, 30}] (* Robert A. Russell, Oct 14 2018 *)
Ach[n_, k_] := Ach[n, k] = If[n < 2, Boole[n == k && n >= 0], k Ach[n-2, k] + Ach[n-2, k-1] + Ach[n-2, k-2]]
k=5; Table[(StirlingS2[n, k] + Ach[n, k])/2, {n, 1, 30}] (* Robert A. Russell, Oct 14 2018 *)
LinearRecurrence[{13, -48, -36, 551, -683, -1542, 3546, 80, -4280, 2400}, {0, 0, 0, 0, 1, 9, 76, 542, 3523, 21393}, 30] (* Robert A. Russell, Oct 14 2018 *)

Crossrefs

Column 5 of A284949.

Cf. A056312.

Cf. A000481 (oriented), A320528 (chiral), A304975 (achiral).

Sequence in context: A255931 A319957 A056339 * A190982 A082677 A185818

Adjacent sequences: A056326 A056327 A056328 * A056330 A056331 A056332

Keyword

nonn

Author

Marks R. Nester

Status

approved