A368518 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 + 3*x^2.

1, 1, 2, 2, 4, 7, 3, 10, 18, 20, 5, 20, 51, 68, 61, 8, 40, 118, 220, 251, 182, 13, 76, 264, 584, 905, 888, 547, 21, 142, 558, 1452, 2678, 3540, 3076, 1640, 34, 260, 1145, 3380, 7279, 11536, 13418, 10456, 4921, 55, 470, 2286, 7548, 18391, 33990, 47600, 49552

Offset: 1

Author

Clark Kimberling, Jan 22 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    2
   2    4    7
   3   10   18    20
   5   20   51    68    61
   8   40  118   220   251   182
  13   76  264   584   905   888   547
  21  142  558  1452  2678  3540  3076  1640
Row 4 represents the polynomial p(4,x) = 3 + 10*x + 18*x^2 + 20*x^3, so (T(4,k)) = (3,10,18,20), k=0..3.
		

Links

• Rigoberto Flórez, Robinson A. Higuita, and Antara Mikherjee, Characterization of the strong divisibility property for generalized Fibonacci polynomials, Integers 18 (2018) 1-28.

Crossrefs

Cf. A000045 (column 1); A002605, (p(n,n-1)); A030195 (row sums), (p(n,1)); A182228 (alternating row sums), (p(n,-1)); A015545, (p(n,2)); A099012, (p(n,-2)); A087567, (p(n,3)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300, A367301, A368150, A368151, A368152, A368153, A368154, A368155, A368156.

Programs

Mathematica

p[1, x_] := 1; p[2, x_] := 1 + 2 x; u[x_] := p[2, x]; v[x_] := 1 + 3x^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 + 32*x^2.

p(n,x) = k*(b^n - c^n), where k = -1/sqrt(5 + 4*x + 16*x^2), b = (1/2)*(2*x + 1 - 1/k), c = (1/2)*(2*x + 1 + 1/k).