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A367300
Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 3 + 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 - 2*x - x^2.
16
1, 3, 2, 10, 10, 3, 33, 46, 22, 4, 109, 194, 131, 40, 5, 360, 780, 678, 296, 65, 6, 1189, 3036, 3228, 1828, 581, 98, 7, 3927, 11546, 14514, 10100, 4194, 1036, 140, 8, 12970, 43150, 62601, 51664, 26479, 8604, 1722, 192, 9, 42837, 159082, 261598, 249720
OFFSET
1,2
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
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) = 3 + 2*x, u = p(2,x), and v = 1 - 2*x - x^2.
p(n,x) = k*(b^n - c^n), where k = -(1/sqrt(13 + 4*x)), b = (1/2) (2*x + 3 + 1/k), c = (1/2) (2*x + 3 - 1/k).
EXAMPLE
First eight rows:
1 3 2
10 10 3
33 46 22 4
109 194 131 40 5
360 780 678 296 65 6
1189 3036 3228 1828 581 98 7
3927 11546 14514 10100 4194 1036 140 8
Row 4 represents the polynomial p(4,x) = 33 + 46*x + 22*x^2 + 4*x^3, so (T(4,k)) = (33,46,22,4), k=0..3.
MATHEMATICA
p[1, x_] := 1; p[2, x_] := 3 + 2 x; u[x_] := p[2, x]; v[x_] := 1 - 2 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. A006190 (column 1); A000027 (p(n,n-1)); A107839 (row sums, (p(n,1)); A001045 (alternating row sums), (p(n,-1)); A030240 (p(n,2)); A039834 (signed Fibonacci numbers), (p(n,-2)); A016130 (p(n,3)); A225883 (p(n,-3)); A099450 (p(n,-4)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299.
Sequence in context: A302846 A214844 A214966 * A103245 A019242 A064367
KEYWORD
nonn,tabf
AUTHOR
Clark Kimberling, Dec 23 2023
STATUS
approved