A367208 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 - x - x^2.

1, 1, 3, 2, 5, 8, 3, 13, 19, 21, 5, 25, 59, 65, 55, 8, 50, 137, 231, 210, 144, 13, 94, 316, 623, 834, 654, 377, 21, 175, 677, 1615, 2545, 2859, 1985, 987, 34, 319, 1411, 3859, 7285, 9691, 9451, 5911, 2584, 55, 575, 2849, 8855, 19115, 30245, 35105, 30407, 17345, 6765

Offset: 1

Author

Clark Kimberling, Nov 13 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 ten rows:
   1
   1    3
   2    5     8
   3   13    19    21
   5   25    59    65     55
   8   50   137   231    210    144
  13   94   316   623    834    654    377
  21  175   677  1615   2545   2859   1985    987
  34  319  1411  3859   7285   9691   9451   5911   2584
  55  575  2849  8855  19115  30245  35105  30407  17345  6765
Row 4 represents the polynomial p(4,x) = 3 + 13*x + 19*x^2 + 21*x^3, so (T(4,k)) = (3,13,19,21), 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. A000045 (column 1), A001906 (T(n,n-1)), A001353 (row sums, p(n,1)), A077985 (alternating row sums, p(n,-1)), A190974 (p(n,2)), A004254 (p(n,-2)), A190977 (p(n,-3)), A094440, A367209, A367210, A367211, A367297, A367298, A367299, A367300.

Programs

Mathematica

p[1, x_] := 1; p[2, x_] := 1 + 3 x; u[x_] := p[2, x]; v[x_] := 1 - 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) = 1 + 3*x, u = p(2,x), and v = 1 - x - x^2.

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