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, 8991948337723260, 35983044114659472, 143977928423467440, 576048972752188260, 2304607666801990188, 9219666007300387200, 36882370043723748480

Offset: 1

Author

Robert A. Russell, Jun 06 2018

Keywords

• nonn
• easy

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.

Examples

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

Crossrefs

Fourth column of A305622.

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

Cf. A000919, A056311.

Programs

Mathematica

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

PARI

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

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.