A367208 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 = 1 - x - x^2.

1, 1, 3, 2, 5, 8, 3, 13, 19, 21, 5, 25, 59, 65, 55, 8, 50, 137, 231, 210, 144, 13, 94, 316, 623, 834, 654, 377, 21, 175, 677, 1615, 2545, 2859, 1985, 987, 34, 319, 1411, 3859, 7285, 9691, 9451, 5911, 2584, 55, 575, 2849, 8855, 19115, 30245, 35105, 30407

Offset

1,3

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.

Links

Table of n, a(n) for n=1..53.

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

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 = 1 - x - x^2.

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

Example

First ten rows:
  1
  1    3
  2    5     8
  3   13    19    21
  5   25    59    65     55
  8   50   137   231    210    144
 13   94   316   623    834    654    377
 21  175   677  1615   2545   2859   1985    987
 34  319  1411  3859   7285   9691   9451   5911   2584
 55  575  2849  8855  19115  30245  35105  30407  17345  6765
Row 4 represents the polynomial p(4,x) = 3 + 13*x + 19*x^2 + 21*x^3, so (T(4,k)) = (3,13,19,21), k=0..3.

Mathematica

p[1, x_] := 1; p[2, x_] := 1 + 3 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}]]

Crossrefs

Cf. A000045 (column 1), A001906 (T(n,n-1)), A001353 (row sums, p(n,1)), A088985 (alternating row sums, (p(n,-1)), A190974 (p(n,2)), A004254 (p(n,-2)), A190977 ((p,n,-3)), A094440, A367209, A367210, A367211, A367297, A367298, A367299, A367300.

Sequence in context: A124422 A223544 A132776 * A249741 A246275 A209757

Adjacent sequences: A367205 A367206 A367207 * A367209 A367210 A367211

Keyword

nonn,tabl

Author

Clark Kimberling, Nov 13 2023

Status

approved