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A305622 Triangle read by rows: T(n,k) is the number of chiral pairs of rows of n colors with exactly k different colors.

%I A305622 #25 Sep 14 2019 12:20:24
%S A305622 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 A305622 4188,8400,7560,2520,0,120,2880,20400,63000,95760,70560,20160,0,240,
%U A305622 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 A305622 Triangle read by rows: T(n,k) is the number of chiral pairs of rows of n colors with exactly k different colors.
%C A305622 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 A305622 Andrew Howroyd, <a href="/A305622/b305622.txt">Table of n, a(n) for n = 1..1275</a> (first 50 rows)
%F A305622 T(n,k) = (k!/2) * (S2(n,k) - S2(ceiling(n/2),k)) where S2(n,k) is the Stirling subset number A008277.
%F A305622 T(n,k) = (A019538(n,k) - A019538(ceiling(n/2),k)) / 2.
%F A305622 T(n,k) = A019538(n,k) - A305621(n,k).
%F A305622 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 A305622 T(n, k) = Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*A293500(n, i). - _Andrew Howroyd_, Sep 13 2019
%e A305622 The triangle begins:
%e A305622   0;
%e A305622   0,   1;
%e A305622   0,   2,     3;
%e A305622   0,   6,    18,     12;
%e A305622   0,  12,    72,    120,      60;
%e A305622   0,  28,   267,    780,     900,     360;
%e A305622   0,  56,   885,   4188,    8400,    7560,     2520;
%e A305622   0, 120,  2880,  20400,   63000,   95760,    70560,    20160;
%e A305622   0, 240,  9000,  93120,  417000,  952560,  1164240,   725760,  181440;
%e A305622   ...
%e A305622 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 A305622 with(combinat):
%p A305622 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 A305622 Table[(k!/2) (StirlingS2[n, k] - StirlingS2[Ceiling[n/2], k]), {n, 1, 15}, {k, 1, n}] // Flatten
%o A305622 (PARI) T(n,k) = (k!/2) * (stirling(n,k,2) - stirling(ceil(n/2),k,2));
%o A305622 for (n=1, 10, for(k=1, n, print1(T(n,k), ", ")); print); \\ _Michel Marcus_, Sep 27 2018
%Y A305622 Columns 1-6 are A000004, A122746(n-2), A305623, A305624, A305625, and A305626.
%Y A305622 Row sums are A327091.
%Y A305622 Cf. A008277, A019538, A293500, A305621.
%K A305622 nonn,tabl,easy
%O A305622 1,5
%A A305622 _Robert A. Russell_, Jun 06 2018