A367209 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 4*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, 4, 2, 7, 15, 3, 18, 38, 56, 5, 35, 116, 186, 209, 8, 70, 273, 650, 859, 780, 13, 132, 629, 1777, 3366, 3821, 2911, 21, 246, 1352, 4600, 10410, 16556, 16556, 10864, 34, 449, 2820, 11024, 29770, 56874, 78504, 70356, 40545, 55, 810, 5701, 25306, 78324

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..50.

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 + 4*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 + 4*x - D), c = (1/2)*(1 + 4*x + D), where D = sqrt(5 + 4*x + 12*x^2).

Example

First nine rows:
   1
   1    4
   2    7    15
   3   18    38     56
   5   35   116    186    209
   8   70   273    650    859    780
  13  132   629   1777   3366   3821   2911
  21  246  1352   4600  10410  16556  16556  10864
  34  449  2820  11024  29770  56874  78504  70356  405459
Row 4 represents the polynomial p(4,x) = 3 + 18*x + 38*x^2 + 56*x^3, so (T(4,k)) = (3,18,38,56), k=0..3.

Mathematica

p[1, x_] := 1; p[2, x_] := 1 + 4 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), A001353 (T(n,n-1), A004254 (row sums, p(n,1), A006190) (alternating row sums, p(n,-1), A094440, A367208, A367210, A367211, A367297, A367298, A367299, A367300.

Sequence in context: A261690 A211941 A050105 * A348873 A128333 A201414

Adjacent sequences: A367206 A367207 A367208 * A367210 A367211 A367212

Keyword

nonn,tabl

Author

Clark Kimberling, Nov 13 2023

Status

approved