A305625 Number of chiral pairs of rows of n colors with exactly 5 different colors.

0, 0, 0, 0, 60, 900, 8400, 63000, 417000, 2551440, 14802900, 82763100, 450501660, 2404493700, 12645952200, 65771370000, 339164682000, 1737485315640, 8855354531100, 44952362878500, 227475739300260, 1148269299919500, 5785013208282000, 29100046926951000, 146201097996135000, 733811769167043840, 3680292427100043300, 18446421887430345900, 92412024657725026860, 462780012983867889300, 2316780309783100387800

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(5) = 60, the chiral pairs are the 5! = 120 permutations of ABCDE, each paired with its reverse.
		

Crossrefs

Fifth column of A305622.

A056456(n) is number of achiral rows of n colors with exactly 5 different colors.

Cf. A001118, A056312.

Programs

Mathematica

k=5; Table[(k!/2) (StirlingS2[n,k] - StirlingS2[Ceiling[n/2],k]), {n, 1, 40}]

PARI

a(n) = 60*(stirling(n, 5, 2) - stirling(ceil(n/2), 5, 2)); \\ Altug Alkan, Sep 26 2018

Formula

a(n) = (k!/2) * (S2(n,k) - S2(ceiling(n/2),k)), with k=5 colors used and where S2(n,k) is the Stirling subset number A008277.

a(n) = (A001118(n) - A056456(n)) / 2.

a(n) = A001118(n) - A056312(n) = A056312(n) - A056456(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=5 colors used.