%I #13 Dec 25 2023 13:47:10
%S 1,2,5,5,18,24,12,62,126,115,29,192,545,794,551,70,567,2040,4114,4716,
%T 2640,169,1618,7047,17940,28420,26964,12649,408,4508,23020,70582,
%U 140988,185122,150122,60605,985,12336,72222,258492,620379,1027368,1156155,819558
%N Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 2 + 5*x, p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >= 3, where u = p(2,x), v = 1 - 2*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), 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) = 2 + 5*x, u = p(2,x), and v = 1 - 2*x - x^2.
%F p(n,x) = k*(b^n - c^n), where k = -(1/sqrt(8 + 12*x + 21*x^2)), b = (1/2) (5*x + 2 + 1/k), c = (1/2) (5*x + 2 - 1/k).
%e First eight rows:
%e 1
%e 2 5
%e 5 18 24
%e 12 62 126 115
%e 29 192 545 794 551
%e 70 567 2040 4114 4716 2640
%e 169 1618 7047 17940 28420 26964 12649
%e 408 4508 23020 70582 140988 185122 150122 60605
%e Row 4 represents the polynomial p(4,x) = 12 + 62*x + 126*x^2 + 115*x^3, so (T(4,k)) = (12,62,126,115), k=0..3.
%t p[1, x_] := 1; p[2, x_] := 2 + 5 x; u[x_] := p[2, x]; v[x_] := 1 - 2 x - x^2;
%t p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
%t Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
%t Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
%Y Cf. A000129 (column 1); A004254 (p(n,n-1)); A186446 (row sums, (p(n,1)); A007482 (alternating row sums), (p(n,-1)); A041025 (p(n,-2)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367300.
%K nonn,tabl
%O 1,2
%A _Clark Kimberling_, Dec 23 2023