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A368153 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 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 - 3*x - x^2.

%I A368153 #10 Jan 22 2024 06:03:44
%S A368153 1,1,2,2,1,3,3,4,-2,4,5,5,4,-10,5,8,10,-3,4,-25,6,13,16,1,-29,14,-49,
%T A368153 7,21,28,-8,-24,-78,56,-84,8,34,47,-12,-88,-26,-162,168,-132,9,55,80,
%U A368153 -31,-140,-200,100,-330,408,-195,10,89,135,-58,-301,-230,-296
%N A368153 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 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 - 3*x - x^2.
%C A368153 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 A368153 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), 1-28, Paper No. A14.
%F A368153 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 + 2*x, u = p(2,x), and v = 1 - 3*x - x^2.
%F A368153 p(n,x) = k*(b^n - c^n), where k = -1/sqrt(5 - 8*x), b = (1/2)*(2*x + 1 - 1/k), c = (1/2)*(2*x + 1 + 1/k).
%e A368153 First eight rows:
%e A368153    1
%e A368153    1   2
%e A368153    2   1   3
%e A368153    3   4  -2    4
%e A368153    5   5   4  -10    5
%e A368153    8  10  -3    4  -25    6
%e A368153   13  16   1  -29   14  -49    7
%e A368153   21  28  -8  -24  -78   56  -84   8
%e A368153 Row 4 represents the polynomial p(4,x) = 3 + 4*x - 2*x^2 + 4*x^3, so (T(4,k)) = (3,4,-2,4), k=0..3.
%t A368153 p[1, x_] := 1; p[2, x_] := 1 + 2 x; u[x_] := p[2, x]; v[x_] := 1 - 3x - x^2;
%t A368153 p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
%t A368153 Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
%t A368153 Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
%Y A368153 Cf. A000045 (column 1); A000027 (p(n,n-1)); A057083 (row sums), (p(n,1)); A182228 (alternating row sums), (p(n,-1)); A190970, (p(n,2)); A030195, (p(n,-2)); A052918, (p(n,-3)); A190972, (p(n,-4)); A057085, (p(n,-5)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300, A367301, A368150, A368151, A368152.
%K A368153 sign,tabl
%O A368153 1,3
%A A368153 _Clark Kimberling_, Jan 20 2024