A305623 -id:A305623 - cp's OEIS frontend

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

0, 0, 1, 0, 2, 3, 0, 6, 18, 12, 0, 12, 72, 120, 60, 0, 28, 267, 780, 900, 360, 0, 56, 885, 4188, 8400, 7560, 2520, 0, 120, 2880, 20400, 63000, 95760, 70560, 20160, 0, 240, 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

Offset: 1

Robert A. Russell, Jun 06 2018

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.

			The triangle begins:
  0;
  0,   1;
  0,   2,     3;
  0,   6,    18,     12;
  0,  12,    72,    120,      60;
  0,  28,   267,    780,     900,     360;
  0,  56,   885,   4188,    8400,    7560,     2520;
  0, 120,  2880,  20400,   63000,   95760,    70560,    20160;
  0, 240,  9000,  93120,  417000,  952560,  1164240,   725760,  181440;
  ...
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.

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)

Columns 1-6 are A000004, A122746(n-2), A305623, A305624, A305625, and A305626.
Row sums are A327091.
Cf. A008277, A019538, A293500, A305621.

with(combinat):
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

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

T(n,k) = (k!/2) * (stirling(n,k,2) - stirling(ceil(n/2),k,2));
for (n=1, 10, for(k=1, n, print1(T(n,k), ", ")); print); \\ Michel Marcus, Sep 27 2018

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.
T(n,k) = A019538(n,k) - A305621(n,k).
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
T(n, k) = Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*A293500(n, i). - Andrew Howroyd, Sep 13 2019

A056310 Number of reversible strings with n beads using exactly three different colors.

Original entry on oeis.org

0, 0, 3, 18, 78, 273, 921, 2916, 9150, 28065, 85773, 259848, 785778, 2367813, 7128201, 21427956, 64382550, 193326105, 580372293, 1741847328, 5227116378, 15684323853, 47059266081, 141189861996

Offset: 1

Marks R. Nester

A string and its reverse are considered to be equivalent.

			For n=3, the three rows are ABC, ACB, and BAC, being respectively equivalent to CBA, BCA, and CAB, with which they form chiral pairs. - _Robert A. Russell_, Sep 25 2018

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-6,-24,49,6,-66,36).

Cf. A032120, A005418.
Column 3 of A305621.
Equals (A001117 + A056454) / 2 = A001117 - A305623 = A305623 + A056454.

seq(coeff(series(-3*x^3*(12*x^4-5*x^3-4*x^2+1)/((x-1)*(2*x-1)*(3*x-1)*(2*x^2-1)*(3*x^2-1)),x,n+1), x, n), n = 1..25); # Muniru A Asiru, Sep 27 2018

k=3; Table[(StirlingS2[i,k]+StirlingS2[Ceiling[i/2],k])k!/2,{i,k,30}] (* Robert A. Russell, Nov 25 2017 *)
LinearRecurrence[{6, -6, -24, 49, 6, -66, 36}, {0, 0, 3, 18, 78, 273, 921}, 40] (* Vincenzo Librandi, Sep 27 2018 *)

a(n) = A032120(n) - 3*A005418(n+1) + 3.

G.f.: -3*x^3*(12*x^4 - 5*x^3 - 4*x^2 + 1)/((x - 1)*(2*x - 1)*(3*x - 1)*(2*x^2 - 1)*(3*x^2 - 1)). [Colin Barker, Jul 07 2012]

cp's OEIS Frontend

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

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