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 A368150 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^2. 8
 1, 1, 3, 2, 6, 8, 3, 15, 25, 21, 5, 30, 76, 90, 55, 8, 60, 188, 324, 300, 144, 13, 114, 439, 948, 1251, 954, 377, 21, 213, 961, 2529, 4207, 4527, 2939, 987, 34, 390, 2026, 6246, 12606, 17154, 15646, 8850, 2584, 55, 705, 4136, 14640, 34590, 56970, 65840 (list; table; graph; refs; listen; history; text; internal format)
 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..52. 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^2. p(n,x) = k*(b^n - c^n), where k = -1/sqrt(5 + 6*x + 5*x^2), b = (1/2)*(3*x + 1 - 1/k), c = (1/2)*(3*x + 1 + 1/k). EXAMPLE First eight rows: 1 1 3 2 6 8 3 15 25 21 5 30 76 90 55 8 60 188 324 300 144 13 114 439 948 1251 954 377 21 213 961 2529 4207 4527 2939 987 Row 4 represents the polynomial p(4,x) = 3 + 15*x + 25*x^2 + 21*x^3, so (T(4,k)) = (3,15,25,21), k=0..3. MATHEMATICA p[1, x_] := 1; p[2, x_] := 1 + 3 x; u[x_] := p[2, x]; v[x_] := 1 - 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 (p(n,n-1)); A000302 (row sums), (p(n,1)); A122803 (alternating row sums), (p(n,-1)); A190972 (p(n,2)), A116415, (p(n,-2)); A190990, (p(n,3)); A057084, (p(n,-3)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300, A367301, A368151. Sequence in context: A210236 A193998 A209171 * A348686 A160855 A120232 Adjacent sequences: A368147 A368148 A368149 * A368151 A368152 A368153 KEYWORD nonn,tabl AUTHOR Clark Kimberling, Dec 25 2023 STATUS approved

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Last modified June 14 05:27 EDT 2024. Contains 373393 sequences. (Running on oeis4.)