A305623 Number of chiral pairs of rows of n colors with exactly 3 different colors.

0, 0, 3, 18, 72, 267, 885, 2880, 9000, 27915, 85233, 259308, 783972, 2366007, 7122405, 21422160, 64364400, 193307955, 580316313, 1741791348, 5226945372, 15684152847, 47058746925, 141189342840, 423593188200, 1270831465995, 3812595048993, 11437991207388, 34314376250772, 102943948309287, 308833455491445, 926503630549920, 2779517334002400, 8338565015656035, 25015720816575273, 75047214375967428

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(3) = 3, the chiral pairs are ABC-CBA, ACB-BCA, and BAC-CAB.
		

Links

• Simon Plouffe, Conjectures of the OEIS, as of June 20, 2018.

Crossrefs

Third column of A305622.

A056454(n) is number of achiral rows of n colors with exactly 3 different colors.

Cf. A001117, A056310.

Programs

Mathematica

k=3; Table[(k!/2) (StirlingS2[n,k] - StirlingS2[Ceiling[n/2],k]), {n, 1, 40}]

PARI

a(n) = 3*(stirling(n,3,2)-stirling(ceil(n/2),3,2)); \\ Altug Alkan, Sep 26 2018

Formula

a(n) = (k!/2) * (S2(n,k) - S2(ceiling(n/2),k)), with k=3 colors used and where S2(n,k) is the Stirling subset number A008277.

a(n) = (A001117(n) - A056454(n)) / 2.

a(n) = A001117(n) - A056310(n) = A056310(n) - A056454(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=3 colors used.

G.f.: 3*x^3*(5*x^2-x-1)/(-36*x^6+30*x^5+24*x^4-25*x^3-x^2+5*x-1). - Simon Plouffe, Jun 20 2018