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.

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

Andrew Howroyd, Table of n, a(n) for n = 1..1275

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.

Cf. A277504, A305622.

Sequence in context: A098458 A165914 A139621 * A196841 A165732 A197698

Adjacent sequences: A305618 A305619 A305620 * A305622 A305623 A305624

Keyword

nonn,tabl,easy

Author

Robert A. Russell, Jun 06 2018

Status

approved