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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.

%I A367297 #20 Mar 28 2025 04:35:32
%S A367297 1,2,3,5,10,8,12,34,38,21,29,104,161,130,55,70,305,592,654,420,144,
%T A367297 169,866,2023,2788,2436,1308,377,408,2404,6556,10810,11756,8574,3970,
%U A367297 987,985,6560,20446,39164,50779,46064,28987,11822,2584,2378,17663,61912,134960,202630,218717,171232,95078,34690,6765
%N 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.
%C A367297 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 A367297 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 A367297 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.
%F A367297 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).
%e A367297 First eight rows:
%e A367297     1
%e A367297     2    3
%e A367297     5   10    8
%e A367297    12   34   38    21
%e A367297    29  104  161   130    55
%e A367297    70  305  592   654   420  144
%e A367297   169  866 2023  2788  2436 1308  377
%e A367297   408 2404 6556 10810 11756 8574 3970 987
%e A367297 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.
%t A367297 p[1, x_] := 1; p[2, x_] := 2 + 3 x; u[x_] := p[2, x]; v[x_] := 1 - 2 x - x^2;
%t A367297 p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
%t A367297 Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
%t A367297 Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
%Y A367297 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.
%K A367297 nonn,tabl
%O A367297 1,2
%A A367297 _Clark Kimberling_, Nov 26 2023