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A368150 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 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 - x^2.

%I A368150 #8 Dec 30 2023 23:41:06
%S A368150 1,1,3,2,6,8,3,15,25,21,5,30,76,90,55,8,60,188,324,300,144,13,114,439,
%T A368150 948,1251,954,377,21,213,961,2529,4207,4527,2939,987,34,390,2026,6246,
%U A368150 12606,17154,15646,8850,2584,55,705,4136,14640,34590,56970,65840
%N A368150 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 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 - x^2.
%C A368150 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 A368150 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 A368150 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 + 3*x, u = p(2,x), and v = 1 - x^2.
%F A368150 p(n,x) = k*(b^n - c^n), where k = -1/sqrt(5 + 6*x + 5*x^2), b = (1/2)*(3*x + 1 - 1/k), c = (1/2)*(3*x + 1 + 1/k).
%e A368150 First eight rows:
%e A368150    1
%e A368150    1    3
%e A368150    2    6    8
%e A368150    3   15   25    21
%e A368150    5   30   76    90     55
%e A368150    8   60  188   324    300   144
%e A368150   13  114  439   948   1251   954   377
%e A368150   21  213  961  2529   4207  4527  2939   987
%e A368150 Row 4 represents the polynomial p(4,x) = 3 + 15*x + 25*x^2 + 21*x^3, so (T(4,k)) = (3,15,25,21), k=0..3.
%t A368150 p[1, x_] := 1; p[2, x_] := 1 + 3 x; u[x_] := p[2, x]; v[x_] := 1 - x^2;
%t A368150 p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
%t A368150 Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
%t A368150 Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
%Y A368150 Cf. A000045 (column 1); A001906 (p(n,n-1)); A000302 (row sums), (p(n,1)); A122803 (alternating row sums), (p(n,-1)); A190972 (p(n,2)), A116415, (p(n,-2)); A190990, (p(n,3)); A057084, (p(n,-3)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300, A367301, A368151.
%K A368150 nonn,tabl
%O A368150 1,3
%A A368150 _Clark Kimberling_, Dec 25 2023