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A367299 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 2 + 5*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.

%I A367299 #18 Mar 28 2025 04:35:50
%S A367299 1,2,5,5,18,24,12,62,126,115,29,192,545,794,551,70,567,2040,4114,4716,
%T A367299 2640,169,1618,7047,17940,28420,26964,12649,408,4508,23020,70582,
%U A367299 140988,185122,150122,60605,985,12336,72222,258492,620379,1027368,1156155,819558,290376
%N A367299 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 2 + 5*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.
%C A367299 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.
%H A367299 Rigoberto Flórez, Robinson Higuita, and Antara Mukherjee, <a href="http://math.colgate.edu/~integers/s14/s14.Abstract.html">Characterization of the strong divisibility property for generalized Fibonacci polynomials</a>, Integers, 18 (2018), Paper No. A14.
%F A367299 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 + 5*x, u = p(2,x), and v = 1 - 2*x - x^2.
%F A367299 p(n,x) = k*(b^n - c^n), where k = -(1/sqrt(8 + 12*x + 21*x^2)), b = (1/2) (5*x + 2 + 1/k), c = (1/2) (5*x + 2 - 1/k).
%e A367299 First eight rows:
%e A367299     1
%e A367299     2    5
%e A367299     5   18    24
%e A367299    12   62   126   115
%e A367299    29  192   545   794    551
%e A367299    70  567  2040  4114   4716   2640
%e A367299   169 1618  7047 17940  28420  26964  12649
%e A367299   408 4508 23020 70582 140988 185122 150122 60605
%e A367299 Row 4 represents the polynomial p(4,x) = 12 + 62*x + 126*x^2 + 115*x^3, so (T(4,k)) = (12,62,126,115), k=0..3.
%t A367299 p[1, x_] := 1; p[2, x_] := 2 + 5 x; u[x_] := p[2, x]; v[x_] := 1 - 2 x - x^2;
%t A367299 p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
%t A367299 Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
%t A367299 Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
%Y A367299 Cf. A000129 (column 1); A004254 (p(n,n-1)); A186446 (row sums, p(n,1)); A007482 (alternating row sums, p(n,-1)); A041025 (p(n,-2)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367300.
%K A367299 nonn,tabl
%O A367299 1,2
%A A367299 _Clark Kimberling_, Dec 23 2023