A305626 Number of chiral pairs of rows of n colors with exactly 6 different colors.

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.

Links

Table of n, a(n) for n=1..31.

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.

Cf. A000920, A056313.

Sequence in context: A234550 A024185 A033592 * A056322 A056313 A192829

Adjacent sequences: A305623 A305624 A305625 * A305627 A305628 A305629

Keyword

nonn,easy

Author

Robert A. Russell, Jun 06 2018

Status

approved