A305624 Number of chiral pairs of rows of n colors with exactly 4 different colors.

0, 0, 0, 12, 120, 780, 4188, 20400, 93120, 409140, 1748220, 7337232, 30386160, 124696740, 508250988, 2061566400, 8331954240, 33585590580, 135115594140, 542784981552, 2178107091600, 8733341736900, 34996103558988, 140172672276000, 561255446475360, 2246716252964820

Offset

1,4

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

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (7,-3,-63,82,182,-288,-168,288).

Formula

a(n) = (k!/2) * (S2(n,k) - S2(ceiling(n/2),k)), with k=4 colors used and where S2(n,k) is the Stirling subset number A008277.

a(n) = (A000919(n) - A056455(n)) / 2.

a(n) = A000919(n) - A056311(n) = A056311(n) - A056455(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=4 colors used.

G.f.: 12*x^4*(1 + 3*x - 2*x^2 - 13*x^3)/((1 - 2*x)*(1 + 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 2*x^2)*(1 - 3*x^2)). - Andrew Howroyd, Nov 06 2025

Example

For a(4) = 12, the chiral pairs are the 4! = 24 permutations of ABCD, each paired with its reverse.

Mathematica

k=4; Table[(k!/2) (StirlingS2[n, k] - StirlingS2[Ceiling[n/2], k]), {n, 1, 40}]

Prog

(PARI) a(n) = my(k=4); (k!/2) * (stirling(n, k, 2) - stirling(ceil(n/2), k, 2)); \\ Michel Marcus, Jun 07 2018

Crossrefs

Column 4 of A305622.

A056455(n) is number of achiral rows of n colors with exactly 4 different colors.

Cf. A000919, A056311.

Sequence in context: A001816 A354697 A133386 * A056320 A056311 A009050

Adjacent sequences: A305621 A305622 A305623 * A305625 A305626 A305627

Keyword

nonn,easy

Author

Robert A. Russell, Jun 06 2018

Status

approved