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A305621 Triangle read by rows: T(n,k) is the number of rows of n colors with exactly k different colors counting chiral pairs as equivalent, i.e., the rows are reversible.

%I A305621 #16 Sep 13 2019 15:18:37
%S A305621 1,1,1,1,4,3,1,8,18,12,1,18,78,120,60,1,34,273,780,900,360,1,70,921,
%T A305621 4212,8400,7560,2520,1,134,2916,20424,63000,95760,70560,20160,1,270,
%U A305621 9150,93360,417120,952560,1164240,725760,181440,1,526,28065,409380,2551560,8217720,14817600,15120000,8164800,1814400,1,1054,85773,1749780,14804700,64615680,161247240,239500800,209563200,99792000,19958400
%N A305621 Triangle read by rows: T(n,k) is the number of rows of n colors with exactly k different colors counting chiral pairs as equivalent, i.e., the rows are reversible.
%H A305621 Andrew Howroyd, <a href="/A305621/b305621.txt">Table of n, a(n) for n = 1..1275</a>
%F A305621 T(n,k) = (k!/2) * (S2(n,k) + S2(ceiling(n/2),k)) where S2(n,k) is the Stirling subset number A008277.
%F A305621 T(n,k) = (A019538(n,k) + A019538(ceiling(n/2),k)) / 2.
%F A305621 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 25 2018
%F A305621 T(n, k) = Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*A277504(n, i). - _Andrew Howroyd_, Sep 13 2019
%e A305621 The triangle begins:
%e A305621   1;
%e A305621   1,   1;
%e A305621   1,   4,    3;
%e A305621   1,   8,   18,      12;
%e A305621   1,  18,   78,     120,      60;
%e A305621   1,  34,  273,     780,     900,     360;
%e A305621   1,  70,  921,    4212,    8400,    7560,     2520;
%e A305621   1, 134, 2916,   20424,   63000,   95760,    70560,    20160;
%e A305621   1, 270, 9150,   93360,  417120,  952560,  1164240,   725760,  181440;
%e A305621   ...
%e A305621 For T(3,2)=4, the achiral color rows are ABA and BAB, while the chiral pairs are AAB-BAA and ABB-BBA. For T(3,3)=3, the color rows are all chiral pairs: ABC-CBA, ACB-BCA, and BAC-CAB.
%t A305621 Table[(k!/2) (StirlingS2[n, k] + StirlingS2[Ceiling[n/2], k]), {n, 1, 15}, {k, 1, n}] // Flatten
%o A305621 (PARI) T(n,k) = {k! * (stirling(n,k,2) + stirling((n+1)\2,k,2)) / 2} \\ _Andrew Howroyd_, Sep 13 2019
%Y A305621 Columns 1-6 are A057427, A056309, A056310, A056311, A056312, and A056313.
%Y A305621 Row sums are A326963.
%Y A305621 A019538 counts chiral pairs as two, i.e., the rows are not reversible.
%Y A305621 Cf. A277504, A305622.
%K A305621 nonn,tabl,easy
%O A305621 1,5
%A A305621 _Robert A. Russell_, Jun 06 2018