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A368149 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 - x^2.

%I A368149 #12 Dec 30 2023 23:40:52
%S A368149 1,1,2,2,4,3,3,10,10,4,5,20,31,20,5,8,40,78,76,35,6,13,76,184,232,161,
%T A368149 56,7,21,142,406,636,582,308,84,8,34,260,861,1604,1831,1296,546,120,9,
%U A368149 55,470,1766,3820,5215,4630,2640,912,165,10,89,840,3533,8696
%N A368149 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 - x^2.
%C A368149 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 A368149 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 A368149 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 - x^2.
%F A368149 p(n,x) = k*(b^n - c^n), where k = -1/sqrt(5 + 4*x), b = (1/2)*(2*x + 1 - 1/k), c = (1/2)*(2*x + 1 + 1/k).
%e A368149 First eight rows:
%e A368149    1
%e A368149    1    2
%e A368149    2    4    3
%e A368149    3   10   10    4
%e A368149    5   20   31   20    5
%e A368149    8   40   78   76   35    6
%e A368149   13   76  184  232  161   56   7
%e A368149   21  142  406  636  582  308  84  8
%e A368149 Row 4 represents the polynomial p(4,x) = 3 + 10*x + 10*x^2 + 4*x^3, so (T(4,k)) = (3,10,10,4), k=0..3.
%t A368149 p[1, x_] := 1; p[2, x_] := 1 + 2 x; u[x_] := p[2, x]; v[x_] := 1 - x^2;
%t A368149 p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
%t A368149 Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
%t A368149 Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
%Y A368149 Cf. A000045 (column 1); A000027 (p(n,n-1)); A000244 (row sums), (p(n,1)); A033999 (alternating row sums), (p(n,-1)); A116415 (p(n,2)), A000748, (p(n,-2)); A152268, (p(n,3)); A190969, (p(n,-3)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300, A367301, A368150.
%K A368149 nonn,tabl
%O A368149 1,3
%A A368149 _Clark Kimberling_, Dec 25 2023