OFFSET
1,5
COMMENTS
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.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)
FORMULA
T(n,k) = (k!/2) * (S2(n,k) - S2(ceiling(n/2),k)) where S2(n,k) is the Stirling subset number A008277.
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
EXAMPLE
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.
MAPLE
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
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!/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
CROSSREFS
KEYWORD
AUTHOR
Robert A. Russell, Jun 06 2018
STATUS
approved