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Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 3 + 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.
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%I #15 Dec 25 2023 13:47:18

%S 1,3,3,10,16,8,33,75,63,21,109,320,380,220,55,360,1296,1980,1620,720,

%T 144,1189,5070,9459,9940,6255,2262,377,3927,19353,42615,54561,44085,

%U 22635,6909,987,12970,72532,184034,277480,272854,179972,78230,20672,2584,42837

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

%F p(n,x) = k*(b^n - c^n), where k = -(1/sqrt(13 + 10*x + 5*x^2)), b = (1/2) (3*x + 3 + 1/k), c = (1/2) (3*x + 3 - 1/k).

%e First eight rows:

%e 1

%e 3 3

%e 10 16 8

%e 33 75 63 21

%e 109 320 380 220 55

%e 360 1296 1980 1620 720 144

%e 1189 5070 9459 9940 6255 2262 377

%e 3927 19353 42615 54561 44085 22635 6909 987

%e Row 4 represents the polynomial p(4,x) = 33 + 75*x + 63*x^2 + 21*x^3, so (T(4,k)) = (33,75,63,21), k=0..3.

%t p[1, x_] := 1; p[2, x_] := 3 + 3 x; u[x_] := p[2, x]; v[x_] := 1 - 2 x - x^2;

%t p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]

%t Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]

%t Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]

%Y Cf. A006190 (column 1); A001906 (p(n,n-1)); A154244 (row sums), (p(n,1)); A077957 (alternating row sums), (p(n,-1)); A190984 (p(n,2)); (A006190 signed, (p(n,-2)); A154244 (p(n,-3)); A190984 (p(n,-4)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Dec 23 2023