A305622 - OEIS

%I #25 Sep 14 2019 12:20:24

%S 0,0,1,0,2,3,0,6,18,12,0,12,72,120,60,0,28,267,780,900,360,0,56,885,

%T 4188,8400,7560,2520,0,120,2880,20400,63000,95760,70560,20160,0,240,

%U 9000,93120,417000,952560,1164240,725760,181440,0,496,27915,409140,2551440,8217720,14817600,15120000,8164800,1814400,0,992,85233,1748220,14802900,64614960,161247240,239500800,209563200,99792000,19958400

%N Triangle read by rows: T(n,k) is the number of chiral pairs of rows of n colors with exactly k 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 Andrew Howroyd, <a href="/A305622/b305622.txt">Table of n, a(n) for n = 1..1275</a> (first 50 rows)

%F T(n,k) = (k!/2) * (S2(n,k) - S2(ceiling(n/2),k)) where S2(n,k) is the Stirling subset number A008277.

%F T(n,k) = (A019538(n,k) - A019538(ceiling(n/2),k)) / 2.

%F T(n,k) = A019538(n,k) - A305621(n,k).

%F G.f. for column k: k! x^k / (2*Product_{i=1..k}(1-ix)) - k! (x^(2k-1)+x^(2k)) / (2*Product{i=1..k}(1-i x^2)). - _Robert A. Russell_, Sep 26 2018

%F T(n, k) = Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*A293500(n, i). - _Andrew Howroyd_, Sep 13 2019

%e The triangle begins:

%e   0;

%e   0,   1;

%e   0,   2,     3;

%e   0,   6,    18,     12;

%e   0,  12,    72,    120,      60;

%e   0,  28,   267,    780,     900,     360;

%e   0,  56,   885,   4188,    8400,    7560,     2520;

%e   0, 120,  2880,  20400,   63000,   95760,    70560,    20160;

%e   0, 240,  9000,  93120,  417000,  952560,  1164240,   725760,  181440;

%e   ...

%e For T(3,2)=2, the chiral pairs are AAB-BAA and ABB-BBA.  For T(3,3)=3, the chiral pairs are ABC-CBA, ACB-BCA, and BAC-CAB.

%p with(combinat):

%p a:=(n,k)->(factorial(k)/2)* (Stirling2(n,k)-Stirling2(ceil(n/2),k)): seq(seq(a(n,k),k=1..n),n=1..11); # _Muniru A Asiru_, Sep 27 2018

%t Table[(k!/2) (StirlingS2[n, k] - StirlingS2[Ceiling[n/2], k]), {n, 1, 15}, {k, 1, n}] // Flatten

%o (PARI) T(n,k) = (k!/2) * (stirling(n,k,2) - stirling(ceil(n/2),k,2));

%o for (n=1, 10, for(k=1, n, print1(T(n,k), ", ")); print); \\ _Michel Marcus_, Sep 27 2018

%Y Columns 1-6 are A000004, A122746(n-2), A305623, A305624, A305625, and A305626.

%Y Row sums are A327091.

%Y Cf. A008277, A019538, A293500, A305621.

%K nonn,tabl,easy

%O 1,5

%A _Robert A. Russell_, Jun 06 2018