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A367298 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 2 + 4*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 A367298 #18 Mar 28 2025 04:35:14
%S A367298 1,2,4,5,14,15,12,48,76,56,29,148,326,372,209,70,436,1212,1904,1718,
%T A367298 780,169,1242,4169,8228,10191,7642,2911,408,3456,13576,32176,49992,
%U A367298 51488,33112,10864,985,9448,42492,117304,218254,281976,249612,140712,40545
%N A367298 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 2 + 4*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 A367298 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 A367298 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 A367298 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 + 4*x, u = p(2,x), and v = 1 - 2*x - x^2.
%F A367298 p(n,x) = k*(b^n - c^n), where k = -(1/sqrt(8 + 8*x + 12*x^2)), b = (1/2)*(4*x + 2 + 1/k), c = (1/2)*(4*x + 2 - 1/k).
%e A367298 First eight rows:
%e A367298     1
%e A367298     2     4
%e A367298     5    14     15
%e A367298    12    48     76     56
%e A367298    29   148    326    372    209
%e A367298    70   436   1212   1904   1718   780
%e A367298   169  1242   4169   8228  10191  7642    2911
%e A367298   408  3456  13576  32176  49992  51488  33112  10864
%e A367298 Row 4 represents the polynomial p(4,x) = 12 + 48*x + 76*x^2 + 56*x^3, so (T(4,k)) = (12,48,76,56), k=0..3.
%t A367298 p[1, x_] := 1; p[2, x_] := 2 + 4 x; u[x_] := p[2, x]; v[x_] := 1 - 2 x - x^2;
%t A367298 p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
%t A367298 Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
%t A367298 Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
%Y A367298 Cf. A000129 (column 1), A001353 (p(n,n-1)), A154244 (row sums, p(n,1)), A002605 (alternating row sums, p(n,-1)), A190989 (p(n,2)), A005668 (p(n,-2)), A190869 (p(n,-3)), A094440, A367208, A367209, A367210, A367211, A367297, A367299, A367300, A367301.
%K A367298 nonn,tabl
%O A367298 1,2
%A A367298 _Clark Kimberling_, Nov 26 2023