A368149 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 2*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^2.

1, 1, 2, 2, 4, 3, 3, 10, 10, 4, 5, 20, 31, 20, 5, 8, 40, 78, 76, 35, 6, 13, 76, 184, 232, 161, 56, 7, 21, 142, 406, 636, 582, 308, 84, 8, 34, 260, 861, 1604, 1831, 1296, 546, 120, 9, 55, 470, 1766, 3820, 5215, 4630, 2640, 912, 165, 10, 89, 840, 3533, 8696

Offset: 1

Author

Clark Kimberling, Dec 25 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
   1    2
   2    4    3
   3   10   10    4
   5   20   31   20    5
   8   40   78   76   35    6
  13   76  184  232  161   56   7
  21  142  406  636  582  308  84  8
Row 4 represents the polynomial p(4,x) = 3 + 10*x + 10*x^2 + 4*x^3, so (T(4,k)) = (3,10,10,4), 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); A000027 (p(n,n-1)); A000244 (row sums), (p(n,1)); A033999 (alternating row sums), (p(n,-1)); A116415 (p(n,2)), A000748, (p(n,-2)); A152268, (p(n,3)); A190969, (p(n,-3)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300, A367301, A368150.

Programs

Mathematica

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

p(n,x) = k*(b^n - c^n), where k = -1/sqrt(5 + 4*x), b = (1/2)*(2*x + 1 - 1/k), c = (1/2)*(2*x + 1 + 1/k).