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A305626
Number of chiral pairs of rows of n colors with exactly 6 different colors.
2
0, 0, 0, 0, 0, 360, 7560, 95760, 952560, 8217720, 64614960, 476514360, 3355664760, 22837086720, 151449482520, 984573465120, 6302069010720, 39847409421480, 249509368422720, 1550188394120520, 9570844541994120, 58789922099665680, 359629148397511080, 2192484972513916080, 13329510116645202480, 80854267307329446840, 489528474458978944080, 2959252601445086408280, 17866194139995100525080, 107751636988750077294240, 649286502010403671101240
OFFSET
1,6
COMMENTS
If the row is achiral, i.e., the same as its reverse, we ignore it. If different from its reverse, we count it and its reverse as a chiral pair.
FORMULA
a(n) = (k!/2) * (S2(n,k) - S2(ceiling(n/2),k)), with k=6 colors used and where S2(n,k) is the Stirling subset number A008277.
a(n) = (A000920(n) - A056457(n)) / 2.
a(n) = A000920(n) - A056313(n) = A056313(n) - A056457(n).
G.f.: -(k!/2) * (x^(2k-1) + x^(2k)) / Product_{j=1..k} (1 - j*x^2) + (k!/2) * x^k / Product_{j=1..k} (1 - j*x) with k=6 colors used.
EXAMPLE
For a(6) = 360, the chiral pairs are the 6! = 720 permutations of ABCDEF, each paired with its reverse.
MATHEMATICA
k=6; Table[(k!/2) (StirlingS2[n, k] - StirlingS2[Ceiling[n/2], k]), {n, 1, 40}]
PROG
(PARI) a(n) = 360*(stirling(n, 6, 2) - stirling(ceil(n/2), 6, 2)); \\ Altug Alkan, Sep 26 2018
CROSSREFS
Sixth column of A305622.
A056457(n) is number of achiral rows of n colors with exactly 6 different colors.
Sequence in context: A234550 A024185 A033592 * A056322 A056313 A192829
KEYWORD
nonn,easy
AUTHOR
Robert A. Russell, Jun 06 2018
STATUS
approved