OFFSET
0,13
COMMENTS
Let Z_n={0,1,...,n-1} denote the integers mod n,
let U(n) denote the units mod n, the elements in Z_n relatively prime to n.
Let S and S' be two k-subsets of Z_n.
Define an equivalence relation on the set of k-subsets as follows:
S is (u,z)-equivalent to S' iff there is a u in U(n) and a z in Z_n such that S=uS'+z.
Then define X(n,k) to be the number of such (u,z)-equivalence classes.
This sequence is the `X(n,k)' triangle read by rows.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50).
EXAMPLE
The triangle begins
1;
1,1;
1,1,1;
1,1,1,1;
1,1,2,1,1;
1,1,1,1,1,1;
1,1,3,3,3,1,1;
1,1,1,2,2,1,1,1;
1,1,3,4,6,4,3,1,1;
...
For example row 8 is 1,1,3,4,6,4,3,1,1.
We have X(8,3)=4 because there are 4 (u,z)-equivalence classes of 3-subsets in Z_8, their representatives are: {0,1,2}, {0,1,3}, {0,1,4}, and {0,2,4}.
PROG
(PARI)
Follow(s, f)={my(t=f(s), k=1); while(t>s, k++; t=f(t)); if(s==t, k, 0)}
C(n, k, t, x)=prod(u=0, n-1, my(t=Follow(u, v->(v*k+t)%n)); 1 + if(t, x^t));
row(n)=Vecrev(if(n==0, 1, sum(t=0, n-1, sum(k=1, n, if (gcd(k, n)==1, C(n, k, t, 'x), 0)))/(n * eulerphi(n))));
{ for(n=0, 10, print(row(n))) } \\ Andrew Howroyd, Apr 04 2023
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
John P. McSorley, Dec 07 2009
EXTENSIONS
Offset corrected and terms a(45) and beyond from Andrew Howroyd, Apr 04 2023
STATUS
approved