A368154 - OEIS

%I #10 Jan 22 2024 06:04:05

%S 1,1,3,2,3,8,3,9,7,21,5,15,31,15,55,8,30,53,99,30,144,13,54,124,165,

%T 306,54,377,21,99,241,447,481,927,77,987,34,177,487,909,1509,1341,

%U 2767,33,2584,55,315,941,1995,3135,4905,3605,8163,-355,6765,89,555

%N Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 3*x, p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 1 - 3*x - x^2.

%C Because (p(n,x)) is a strong divisibility sequence, for each integer k, the sequence (p(n,k)) is a strong divisibility sequence of integers.

%H Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, <a href="http://math.colgate.edu/~integers/s14/s14.Abstract.html">Characterization of the strong divisibility property for generalized Fibonacci polynomials</a>, Integers, 18 (2018), 1-28, Paper No. A14.

%F p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where p(1,x) = 1, p(2,x) = 1 + 3*x, u = p(2,x), and v = 1 - 3*x - x^2.

%F p(n,x) = k*(b^n - c^n), where k = -1/sqrt(5 - 6*x + 5*x^2), b = (1/2)*(3*x + 1 - 1/k), c = (1/2)*(2*x + 1 + 1/k).

%e First eight rows:

%e    1

%e    1    3

%e    2    3     8

%e    3    9     7    21

%e    5   15    31    15    55

%e    8   30    53    99    30   144

%e   13   54   124   165   306    54  377

%e   21   99   241   447   481   927   77  987

%K tabl,sign

%O 1,3

%A _Clark Kimberling_, Jan 20 2024