%I #13 Apr 26 2022 10:22:38

%S 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,

%T 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,

%U 2,0,1,1,10,0,54,0,1350,0,52,0,2,0,1

%N 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.

%C 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.

%H Pontus von Brömssen, <a href="/A353433/a353433.svg">Plot of T(n,7) for 0 <= n <= 200</a>

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

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

%F 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):

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

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

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

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

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

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

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

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

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

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

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

%e Array begins:

%e n\m| 1 2 3 4 5 6 7 8 9 10

%e ---+------------------------------------

%e 0 | 1 1 1 1 1 1 1 1 1 1

%e 1 | 1 1 2 2 4 2 6 4 6 4

%e 2 | 1 0 2 0 12 0 30 0 18 0

%e 3 | 1 0 2 0 28 0 126 0 54 0

%e 4 | 1 0 2 0 48 0 444 0 162 0

%e 5 | 1 0 2 0 60 0 1350 0 486 0

%e 6 | 1 0 2 0 60 0 3582 0 1458 0

%e 7 | 1 0 2 0 52 0 8550 0 4374 0

%e 8 | 1 0 2 0 48 0 17364 0 13122 0

%e 9 | 1 0 2 0 48 0 30126 0 39366 0

%e 10 | 1 0 2 0 48 0 44922 0 118098 0

%Y Cf. A353434, A353435.

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

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

%K nonn,tabl

%O 0,8

%A _Pontus von Brömssen_, Apr 21 2022