A367210 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 5x, 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, 5, 2, 9, 24, 3, 23, 63, 115, 5, 45, 191, 397, 551, 8, 90, 453, 1381, 2358, 2640, 13, 170, 1044, 3807, 9226, 13482, 12649, 21, 317, 2249, 9865, 28785, 58513, 75061, 60605, 34, 579, 4695, 23703, 82485, 202887, 357567, 409779, 290376, 55, 1045, 9501

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 eight rows:
 1
 1    5
 2    9    24
 3   23    63   115
 5   45   191   397    551
 8   90   453  1381   2358   2640
13  170  1044  3807   9226  13482  12649
21  317  2249  9865  28785  58513  75061  60605
Row 4 represents the polynomial p(4,x) = 3 + 23 x + 63 x^2 + 115 x^3, so  that (T(4,k)) = (3,23,63,115), 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), A004254 (T(n,n-1)), A001109 (row sums p(n,1)), A001076 (alternating row sums, p(n,-1)), A094440, A367208, A367209, A367211, A367297, A367298, A367299, A367300.

Programs

Mathematica

p[1, x_] := 1; p[2, x_] := 1 + 5 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 + 5x, 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 + 5 x - D), c = 1/2 (1 + 5 x + D), where D = sqrt(5 + 6 x + 21 x^2).