A367299 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.

1, 2, 5, 5, 18, 24, 12, 62, 126, 115, 29, 192, 545, 794, 551, 70, 567, 2040, 4114, 4716, 2640, 169, 1618, 7047, 17940, 28420, 26964, 12649, 408, 4508, 23020, 70582, 140988, 185122, 150122, 60605, 985, 12336, 72222, 258492, 620379, 1027368, 1156155, 819558, 290376

Offset: 1

Author

Clark Kimberling, Dec 23 2023

Keywords

• nonn
• tabl

Comments

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.

Examples

			First eight rows:
    1
    2    5
    5   18    24
   12   62   126   115
   29  192   545   794    551
   70  567  2040  4114   4716   2640
  169 1618  7047 17940  28420  26964  12649
  408 4508 23020 70582 140988 185122 150122 60605
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.
		

Links

• Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), Paper No. A14.

Crossrefs

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.

Programs

Mathematica

p[1, x_] := 1; p[2, x_] := 2 + 5 x; u[x_] := p[2, x]; v[x_] := 1 - 2 x - x^2;
p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]

Formula

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.

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).