A056328 Number of reversible string structures with n beads using exactly four different colors.

0, 0, 0, 1, 6, 37, 183, 877, 3930, 17185, 73095, 306361, 1267266, 5198557, 21182343, 85910917, 347187210, 1399451545, 5629911015, 22616256721, 90754855026, 363890126677, 1458172596903, 5840531635357, 23385650196090

Offset: 1

Author

Marks R. Nester

Keywords

• nonn

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 four different elements. An unoriented row is equivalent to its reverse. - Robert A. Russell, Oct 14 2018

Examples

			For a(5)=6, the color patterns are ABCDA, ABCBD, AABCD, ABACD, ABCAD, and ABBCD. The first two are achiral. - _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

• Index entries for linear recurrences with constant coefficients, signature (8, -12, -44, 121, 12, -228, 144).

Crossrefs

Column 4 of A284949.

Cf. A056311.

Cf. A000453 (oriented), A320527 (chiral), A304974 (achiral).

Programs

Mathematica

k=4; Table[(StirlingS2[n,k] + If[EvenQ[n], StirlingS2[n/2+2,4] - StirlingS2[n/2+1,4] - 2StirlingS2[n/2,4], 2StirlingS2[(n+3)/2,4] - 4StirlingS2[(n+1)/2,4]])/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 = 4; Table[(StirlingS2[n, k] + Ach[n, k])/2, {n, 1, 30}] (* Robert A. Russell, Oct 14 2018 *)
LinearRecurrence[{8, -12, -44, 121, 12, -228, 144}, {0, 0, 0, 1, 6, 37, 183}, 30] (* Robert A. Russell, Oct 14 2018 *)

Formula

a(n) = A056323(n) - A001998(n-1).

Empirical g.f.: -x^4*(3*x^3 + x^2 - 2*x + 1) / ((x-1)*(2*x-1)*(2*x+1)*(3*x-1)*(4*x-1)*(3*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=4 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) = (A000453(n) + A304974(n)) / 2 = A000453(n) - A320527(n) = A320527(n) + A304974(n). (End)