A368156 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, 5, 3, 10, 14, 12, 5, 20, 41, 44, 29, 8, 40, 98, 148, 131, 70, 13, 76, 224, 408, 497, 376, 169, 21, 142, 482, 1044, 1542, 1588, 1052, 408, 34, 260, 1003, 2492, 4351, 5456, 4894, 2888, 985, 55, 470, 2026, 5684, 11359, 16790, 18400, 14672, 7813

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    2
   2    4    5
   3   10   14    12
   5   20   41    44    29
   8   40   98   148   131    70
  13   76  224   408   497   376   169
  21  142  482  1044  1542  1588  1052  408
Row 4 represents the polynomial p(4,x) = 3 + 10*x + 14*x^2 + 12*x^3, so (T(4,k)) = (3,10,14,12), 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); A000129, (p(n,n-1)); A007482 (row sums), (p(n,1)); A077925 (alternating row sums), (p(n,-1)); A057088, (p(n,2)); A015523, (p(n,-2)); A015568, (p(n,3)); A180250, (p(n,-3)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300, A367301, A368150, A368151, A368152, A368153, A368154, A368155.

Programs

Mathematica

p[1, x_] := 1; p[2, x_] := 1 + 2 x; u[x_] := p[2, x]; v[x_] := 1 + 2x^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 + 8*x^2), b = (1/2)*(2*x + 1 - 1/k), c = (1/2)*(2*x + 1 + 1/k).