A320526 a(n) is the number of chiral pairs of color patterns (set partitions) in a row of length n using exactly 3 colors (subsets).

0, 0, 0, 2, 10, 40, 141, 464, 1480, 4600, 14145, 43052, 130480, 393820, 1186521, 3568784, 10725760, 32213200, 96714465, 290284052, 871142800, 2613981700, 7843080201, 23531425304, 70598731840, 211804847800, 635432109585, 1906330676252, 5719061512720, 17157321139180

Offset

1,4

Comments

Two color patterns are equivalent if we permute the colors. Chiral color patterns must not be equivalent if we reverse the order of the pattern.

Links

Table of n, a(n) for n=1..30.

Index entries for linear recurrences with constant coefficients, signature (6, -6, -24, 49, 6, -66, 36).

Formula

a(n) = (S2(n,k) - A(n,k))/2, where k=3 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].

G.f.: (x^3 / Product_{k=1..3} (1 - k*x) - x^3*(1 + 2 x)/((1 - 2 x^2)*(1 - 3 x^2))) / 2.

a(n) = (A000392(n) - A304973(n)) / 2 = A000392(n) - A056327(n) = A056327(n) - A304973(n).

Example

For a(4)=2, the two chiral pairs are AABC-ABCC and ABAC-ABCB.

Mathematica

k=3; Table[(StirlingS2[n, k] - If[EvenQ[n], 2StirlingS2[n/2+1, 3] - 2StirlingS2[n/2, 3], StirlingS2[(n+3)/2, 3] - StirlingS2[(n+1)/2, 3]])/2, {n, 1, 30}]
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]] (* A304972 *)
k = 3; Table[(StirlingS2[n, k] - Ach[n, k])/2, {n, 1, 30}]
LinearRecurrence[{6, -6, -24, 49, 6, -66, 36}, {0, 0, 0, 2, 10, 40,
  141}, 40]

Prog

(PARI) m=40; v=concat([0, 0, 0, 2, 10, 40, 141], vector(m-7)); for(n=8, m, v[n] = 6*v[n-1] -6*v[n-2] -24*v[n-3] +49*v[n-4] +6*v[n-5] -66*v[n-6] +36*v[n-7] ); v \\ G. C. Greubel, Oct 16 2018
(Magma) I:=[0, 0, 0, 2, 10, 40, 141]; [n le 7 select I[n] else 6*Self(n-1) -6*Self(n-2) -24*Self(n-3) +49*Self(n-4) +6*Self(n-5) -66*Self(n-6) +36*Self(n-7): n in [1..40]]; // G. C. Greubel, Oct 16 2018

Crossrefs

Column 3 of A320525.

Cf. A000392 (oriented), A056327 (unoriented), A304973 (achiral).

Sequence in context: A009338 A261473 A174395 * A193519 A268329 A223095

Adjacent sequences: A320523 A320524 A320525 * A320527 A320528 A320529

Keyword

nonn,easy

Author

Robert A. Russell, Oct 14 2018

Status

approved