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A320936
Number of chiral pairs of color patterns (set partitions) for a row of length n using 6 or fewer colors (subsets).
5
0, 0, 1, 4, 20, 86, 409, 1976, 10168, 54208, 299859, 1699012, 9808848, 57335124, 338073107, 2004955824, 11936998016, 71253827696, 426061036747, 2550545918300, 15280090686256, 91588065861292, 549159350303235, 3293482358956552, 19755007003402944
OFFSET
1,4
COMMENTS
Two color patterns are equivalent if the colors are permuted.
A chiral row is not equivalent to its reverse.
There are nonrecursive formulas, generating functions, and computer programs for A056273 and A305752, which can be used in conjunction with the first formula.
LINKS
Index entries for linear recurrences with constant coefficients, signature (16,-84,84,685,-2140,180,7200,-8244, -4176,11664,-5184).
FORMULA
a(n) = (A056273(n) - A305752(n))/2.
a(n) = A056273(n) - A056325(n).
a(n) = A056325(n) - A305752(n).
a(n) = A122746(n-2) + A320526(n) + A320527(n) + A320528(n) + A320529(n).
a(n) = Sum_{j=1..k} (S2(n,j) - Ach(n,j)) / 2, where k=6 is the maximum number of colors, S2 is the Stirling subset number A008277, and Ach(n,k) = [n>=0 & n<2 & n==k] + [n>1]*(k*Ach(n-2,k) + Ach(n-2,k-1) + Ach(n-2,k-2)).
From Colin Barker, Nov 22 2018: (Start)
G.f.: x^3*(1 - 12*x + 40*x^2 + 18*x^3 - 308*x^4 + 376*x^5 + 364*x^6 - 882*x^7 + 378*x^8) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 6*x)*(1 - 2*x^2)*(1 - 3*x^2)*(1 - 6*x^2)).
a(n) = 16*a(n-1) - 84*a(n-2) + 84*a(n-3) + 685*a(n-4) - 2140*a(n-5) + 180*a(n-6) + 7200*a(n-7) - 8244*a(n-8) - 4176*a(n-9) + 11664*a(n-10) - 5184*a(n-11) for n>11.
(End)
EXAMPLE
For a(4)=4, the chiral pairs are AAAB-ABBB, AABA-ABAA, AABC-ABCC, and ABAC-ABCB.
MATHEMATICA
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[Sum[StirlingS2[n, j]-Ach[n, j], {j, k}]/2, {n, 40}]
LinearRecurrence[{16, -84, 84, 685, -2140, 180, 7200, -8244, -4176, 11664, -5184}, {0, 0, 1, 4, 20, 86, 409, 1976, 10168, 54208, 299859}, 40]
PROG
(PARI) concat([0, 0], Vec(x^3*(1 - 12*x + 40*x^2 + 18*x^3 - 308*x^4 + 376*x^5 + 364*x^6 - 882*x^7 + 378*x^8) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 6*x)*(1 - 2*x^2)*(1 - 3*x^2)*(1 - 6*x^2)) + O(x^40))) \\ Colin Barker, Nov 22 2018
CROSSREFS
Column 6 of A320751.
Cf. A056273 (oriented), A056325 (unoriented), A305752 (achiral).
Sequence in context: A250003 A343361 A320935 * A320937 A196953 A093357
KEYWORD
nonn,easy
AUTHOR
Robert A. Russell, Oct 27 2018
STATUS
approved