A305623 - OEIS

%I #31 Sep 27 2018 03:13:18

%S 0,0,3,18,72,267,885,2880,9000,27915,85233,259308,783972,2366007,

%T 7122405,21422160,64364400,193307955,580316313,1741791348,5226945372,

%U 15684152847,47058746925,141189342840,423593188200,1270831465995,3812595048993,11437991207388,34314376250772,102943948309287,308833455491445,926503630549920,2779517334002400,8338565015656035,25015720816575273,75047214375967428

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

%C 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.

%H Simon Plouffe, <a href="http://vixra.org/abs/1409.0048">Conjectures of the OEIS, as of June 20, 2018.</a>

%F 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.

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

%F a(n) = A001117(n) - A056310(n) = A056310(n) - A056454(n).

%F 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.

%F 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

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

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

%o (PARI) a(n) = 3*(stirling(n,3,2)-stirling(ceil(n/2),3,2)); \\ _Altug Alkan_, Sep 26 2018

%Y Third column of A305622.

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

%Y Cf. A001117, A056310.

%K nonn,easy

%O 1,3

%A _Robert A. Russell_, Jun 06 2018