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A367297
Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 2 + 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 - 2*x - x^2.
16
1, 2, 3, 5, 10, 8, 12, 34, 38, 21, 29, 104, 161, 130, 55, 70, 305, 592, 654, 420, 144, 169, 866, 2023, 2788, 2436, 1308, 377, 408, 2404, 6556, 10810, 11756, 8574, 3970, 987, 985, 6560, 20446, 39164, 50779, 46064, 28987, 11822, 2584, 2378, 17663, 61912
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) = 2 + 3*x, u = p(2,x), and v = 1 - 2*x - x^2.
p(n,x) = k*(b^n - c^n), where k = -(1/sqrt(8 + 4*x + 5*x^2)), b = (1/2)*(3*x + 2 + 1/k), c = (1/2)*(3*x + 2 - 1/k).
EXAMPLE
First eight rows:
1
2 3
5 10 8
12 34 38 21
29 104 161 130 55
70 305 592 654 420 144
169 866 2023 2788 2436 1308 377
408 2404 6556 10810 11756 8574 3970 987
Row 4 represents the polynomial p(4,x) = 12 + 34*x + 38*x^2 + 21*x^3, so (T(4,k)) = (12,34,38,21), k=0..3.
MATHEMATICA
p[1, x_] := 1; p[2, x_] := 2 + 3 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. A000129 (column 1), A001906 (p(n,n-1)), A107839 (row sums, (p(n,1)), A077925 (alternating row sums, (p(n,-1)), A023000 (p(n,2)), A001076 (p(n,-2)), A186446 (p(n,-3)), A094440, A367208, A367209, A367210, A367211, A367298, A367299, A367300, A367301.
Sequence in context: A250747 A241262 A244489 * A286144 A038807 A094542
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Nov 26 2023
STATUS
approved