A353433 Array read by descending antidiagonals: T(n,m) is the number of sequences of length n >= 0 with elements in 0..m-1 such that no iterated difference has a common factor with m >= 1.

1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 2, 2, 0, 1, 1, 4, 0, 2, 0, 1, 1, 2, 12, 0, 2, 0, 1, 1, 6, 0, 28, 0, 2, 0, 1, 1, 4, 30, 0, 48, 0, 2, 0, 1, 1, 6, 0, 126, 0, 60, 0, 2, 0, 1, 1, 4, 18, 0, 444, 0, 60, 0, 2, 0, 1, 1, 10, 0, 54, 0, 1350, 0, 52, 0, 2, 0, 1

Offset: 0

Author

Pontus von Brömssen, Apr 21 2022

Keywords

• nonn
• tabl

Comments

T(n,m) is divisible by T(1,m) = A000010(m) if n >= 1, because if r is coprime to m, the sequence (x_1, ..., x_n) satisfies the conditions if and only if the sequence (r*x_1 mod m, ..., r*x_n mod m) does.

Examples

			Array begins:
  n\m| 1  2  3  4  5  6     7  8      9 10
  ---+------------------------------------
   0 | 1  1  1  1  1  1     1  1      1  1
   1 | 1  1  2  2  4  2     6  4      6  4
   2 | 1  0  2  0 12  0    30  0     18  0
   3 | 1  0  2  0 28  0   126  0     54  0
   4 | 1  0  2  0 48  0   444  0    162  0
   5 | 1  0  2  0 60  0  1350  0    486  0
   6 | 1  0  2  0 60  0  3582  0   1458  0
   7 | 1  0  2  0 52  0  8550  0   4374  0
   8 | 1  0  2  0 48  0 17364  0  13122  0
   9 | 1  0  2  0 48  0 30126  0  39366  0
  10 | 1  0  2  0 48  0 44922  0 118098  0
		

Links

• Pontus von Brömssen, Plot of T(n,7) for 0 <= n <= 200

Crossrefs

Cf. A353434, A353435.

Rows: A000012 (n=0), A000010 (n=1), A061780 (every second term of row n=2).

Columns: A000012 (m=1), A019590 (m=2), A040000 (m=3), A130706 (m=4 and m=6).

Formula

For fixed n, T(n,m) is multiplicative with T(n,p^e) = T(n,p)*p^(n*(e-1)).

T(n,m) = A353434(n,m) if m is prime.

For each n >= 0, there exists an n-th degree polynomial P such that T(n,m) = P(m) for sufficiently large primes m. For example (for n >= 4, these are empirical observations only):

T(0,m) = 1 for all m >= 1;

T(1,m) = m-1 for all primes m;

T(2,m) = (m-1)*(m-2) for all primes m;

T(3,m) = (m-1)*(m^2-5*m+7) for primes m >= 3;

T(4,m) = (m-1)*(m^3-9*m^2+30*m-38) for primes m >= 5;

T(5,m) = (m-1)*(m^4-14*m^3+81*m^2-235*m+302) for primes m >= 7;

T(6,m) = (m-1)*(m^5-20*m^4+175*m^3-854*m^2+2401*m-3280) for primes m >= 19.

T(n,2) = 0 for n >= 2.

T(n,3) = 2 for n >= 1.

T(n,5) = 48 for n >= 8.

It appears that T(n,7) = T(n+42,7) for n >= 56. (See linked plot.)