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A320529
Number of chiral pairs of color patterns (set partitions) in a row of length n using exactly 6 colors (subsets).
3
0, 0, 0, 0, 0, 0, 9, 124, 1300, 11316, 89513, 660978, 4658738, 31711638, 210332227, 1367416232, 8752773288, 55343303064, 346540112781, 2153037307846, 13292835205606, 81652655795106, 499484899831775, 3045117929546220, 18513208314957356, 112297592929814292, 679900657841661529, 4110073054119135194, 24814158520762637754
OFFSET
1,7
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
Index entries for linear recurrences with constant coefficients, signature (21, -159, 399, 1085, -8085, 9555, 34125, -98644, 5544, 253764, -248724, -136800, 317520, -129600).
FORMULA
a(n) = (S2(n,k) - A(n,k))/2, where k=6 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^6 / (Product_{k=1..6} (1 - k*x)) - x^6 *(1+x)*(1-4*x^2)*(1+2*x-x^2-4*x^3) / Product_{k=1..6} (1 - k*x^2)) / 2.
a(n) = (A000770(n) - A304976(n)) / 2 = A000770(n) - A056330(n) = A056330(n) - A304976(n).
EXAMPLE
For a(7)=9, the chiral pairs are AABCDEF-ABCDEFF, ABACDEF-ABCDEFE, ABCADEF-ABCDEFD, ABCDAEF-ABCDEFC, ABCDEAF-ABCDEFB, ABBCDEF-ABCDEEF, ABCBDEF-ABCDEDF, ABCDBEF-ABCDECF, and ABCCDEF-ABCDDEF.
MATHEMATICA
k=6; Table[(StirlingS2[n, k] - If[EvenQ[n], StirlingS2[n/2+3, 6] - 3StirlingS2[n/2+2, 6] - 8StirlingS2[n/2+1, 6] + 16StirlingS2[n/2, 6], 3StirlingS2[(n+5)/2, 6] - 17StirlingS2[(n+3)/2, 6] + 20StirlingS2[(n+1)/2, 6]])/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 = 6; Table[(StirlingS2[n, k] - Ach[n, k])/2, {n, 1, 30}]
LinearRecurrence[{21, -159, 399, 1085, -8085, 9555, 34125, -98644, 5544, 253764, -248724, -136800, 317520, -129600}, {0, 0, 0, 0, 0, 0, 9, 124, 1300, 11316, 89513, 660978, 4658738, 31711638}, 30]
PROG
(PARI) x='x+O('x^30); concat(vector(6), Vec((x^6/prod(k=1, 6, 1-k*x) - x^6* (1+x)*(1-4*x^2)*(1+2*x-x^2-4*x^3)/prod(k=1, 6, (1-k*x^2)))/2)) \\ G. C. Greubel, Oct 19 2018
(Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); [0, 0, 0, 0, 0, 0] cat Coefficients(R!((x^6/(&*[1-k*x: k in [1..6]]) - x^6*(1+x)*(1-4*x^2)*(1+2*x-x^2-4*x^3)/(&*[1-k*x^2: k in [1..6]]) )/2)); // G. C. Greubel, Oct 19 2018
CROSSREFS
Column 6 of A320525.
Cf. A000770 (oriented), A056330 (unoriented), A304976 (achiral).
Sequence in context: A211101 A269627 A209504 * A280896 A364940 A138438
KEYWORD
nonn,easy
AUTHOR
Robert A. Russell, Oct 14 2018
STATUS
approved