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A320527 Number of chiral pairs of color patterns (set partitions) in a row of length n using exactly 4 colors (subsets). 5
0, 0, 0, 0, 4, 28, 167, 824, 3840, 16920, 72655, 305140, 1265264, 5193188, 21173607, 85887984, 347150080, 1399355440, 5629755935, 22615859180, 90754215024, 363888497148, 1458169977847, 5840524999144, 23385639542720, 93613165023560, 374664497695215, 1499293455643620, 5999080285068784, 24002040333605908 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

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.

FORMULA

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].

G.f.: (x^4 / Product_{k=1..4} (1 - k*x) - x^4*(1 + x)^2*(1 - 2 x^2) / Product_{k=1..4} (1 - k*x^2)) / 2.

a(n) = (A000453(n) - A304974(n)) / 2 = A000453(n) - A056328(n) = A056328(n) - A304974(n).

EXAMPLE

For a(5)=4, the chiral pairs are AABCD-ABCDD, ABACD-ABCDC, ABBCD-ABCCD and ABCAD-ABCDB.

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}]

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 = 4; Table[(StirlingS2[n, k] - Ach[n, k])/2, {n, 1, 30}]

LinearRecurrence[{8, -12, -44, 121, 12, -228, 144}, {0, 0, 0, 0, 4, 28, 167}, 30]

CROSSREFS

Col. 4 of A320525.

Cf. A000453 (oriented), A056328 (unoriented), A304974 (achiral).

Sequence in context: A272936 A053524 A270943 * A272281 A182190 A317232

Adjacent sequences:  A320524 A320525 A320526 * A320528 A320529 A320530

KEYWORD

nonn,easy

AUTHOR

Robert A. Russell, Oct 14 2018

STATUS

approved

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Last modified October 14 11:49 EDT 2019. Contains 327996 sequences. (Running on oeis4.)