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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.
8
1, 1, 1, 1, 4, 3, 1, 8, 18, 12, 1, 18, 78, 120, 60, 1, 34, 273, 780, 900, 360, 1, 70, 921, 4212, 8400, 7560, 2520, 1, 134, 2916, 20424, 63000, 95760, 70560, 20160, 1, 270, 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
OFFSET
1,5
LINKS
FORMULA
T(n,k) = (k!/2) * (S2(n,k) + S2(ceiling(n/2),k)) where S2(n,k) is the Stirling subset number A008277.
T(n,k) = (A019538(n,k) + A019538(ceiling(n/2),k)) / 2.
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
T(n, k) = Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*A277504(n, i). - Andrew Howroyd, Sep 13 2019
EXAMPLE
The triangle begins:
1;
1, 1;
1, 4, 3;
1, 8, 18, 12;
1, 18, 78, 120, 60;
1, 34, 273, 780, 900, 360;
1, 70, 921, 4212, 8400, 7560, 2520;
1, 134, 2916, 20424, 63000, 95760, 70560, 20160;
1, 270, 9150, 93360, 417120, 952560, 1164240, 725760, 181440;
...
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.
MATHEMATICA
Table[(k!/2) (StirlingS2[n, k] + StirlingS2[Ceiling[n/2], k]), {n, 1, 15}, {k, 1, n}] // Flatten
PROG
(PARI) T(n, k) = {k! * (stirling(n, k, 2) + stirling((n+1)\2, k, 2)) / 2} \\ Andrew Howroyd, Sep 13 2019
CROSSREFS
Columns 1-6 are A057427, A056309, A056310, A056311, A056312, and A056313.
Row sums are A326963.
A019538 counts chiral pairs as two, i.e., the rows are not reversible.
Sequence in context: A098458 A165914 A139621 * A196841 A165732 A197698
KEYWORD
nonn,tabl,easy
AUTHOR
Robert A. Russell, Jun 06 2018
STATUS
approved