A368152 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 = 3 - x^2.

1, 1, 3, 4, 6, 8, 7, 27, 25, 21, 19, 66, 126, 90, 55, 40, 204, 392, 504, 300, 144, 97, 522, 1363, 1884, 1851, 954, 377, 217, 1425, 4065, 7281, 8011, 6435, 2939, 987, 508, 3642, 12332, 24606, 34044, 31446, 21524, 8850, 2584, 1159, 9441, 35236, 82020, 127830

Offset: 1

Author

Clark Kimberling, Jan 20 2024

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
    1    3
    4    6    8
    7   27   25   21
   19   66  126   90   55
   40  204  392  504  300  144
   97  522 1363 1884 1851  954  377
  217 1425 4065 7281 8011 6435 2939 987
Row 4 represents the polynomial p(4,x) = 7 + 27*x + 25*x^2 + 21*x^3, so (T(4,k)) = (7,27,25,21), k=0..3.
		

Links

• Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers, 18 (2018), 1-28, Paper No. A14.

Crossrefs

Cf. A006130 (column 1); A001906 (p(n,n-1)); A090017 (row sums), (p(n,1)); A002605 (alternating row sums), (p(n,-1)); A004187, (p(n,2)); A004254, (p(n,-2)); A190988, (p(n,3)); A190978 (unsigned), (p(n,-3)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300, A367301, A368150, A368151.

Programs

Mathematica

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

p(n,x) = k*(b^n - c^n), where k = -1/sqrt(13 + 6*x + 5*x^2), b = (1/2)*(3*x + 1 - 1/k), c = (1/2)*(3*x + 1 + 1/k).