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A368151 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 3x, p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >=3, where u = p(2,x), v = 2 - x^2.

%I A368151 #6 Jan 20 2024 09:45:54
%S A368151 1,1,3,3,6,8,5,21,25,21,11,48,101,90,55,21,123,290,414,300,144,43,282,
%T A368151 850,1416,1551,954,377,85,657,2255,4671,6109,5481,2939,987,171,1476,
%U A368151 5883,13986,22374,24300,18585,8850,2584,341,3303,14736,40320,74295,97713
%N A368151 Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 1 + 3x, p(n,x) = u*p(n-1,x) + v*p(n-2,x) for n >=3, where u = p(2,x), v = 2 - x^2.
%C A368151 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 A368151 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 A368151 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 = 2 - x^2.
%F A368151 p(n,x) = k*(b^n - c^n), where k = -1/sqrt(9 + 6 x + 5 x^2), b = (1/2) (3 x + 1 - 1/k), c = (1/2) (3 x + 1 + 1/k).
%e A368151 First eight rows:
%e A368151    1
%e A368151    1     3
%e A368151    3     6    8
%e A368151    5    21    25    21
%e A368151   11    48   101    90    55
%e A368151   21   123   290   414   300  144
%e A368151   43   282   850  1416  1551  954    377
%e A368151   85   657  2255  4671  6109  5481  2939  987
%e A368151 Row 4 represents the polynomial p(4,x) = 5 + 21 x + 25 x^2 + 21 x^3, so (T(4,k)) = (5,21,25,21), k=0..3.
%t A368151 p[1, x_] := 1; p[2, x_] := 1 + 3 x; u[x_] := p[2, x]; v[x_] := 2 - x^2;
%t A368151 p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
%t A368151 Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
%t A368151 Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
%Y A368151 Cf. A001045 (column 1); A001906 (p(n,n-1)); A001076 (row sums), (p(n,1)); A077985 (alternating row sums), (p(n,-1)); A186446 (p(n,2)), A107839, (p(n,-2)); A190989, (p(n,3)); A023000, (p(n,-3)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300, A367301, A368150.
%K A368151 nonn,tabl
%O A368151 1,3
%A A368151 _Clark Kimberling_, Dec 31 2023