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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) = 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 - 3*x - 2*x^2.
3

%I #11 Jan 22 2024 00:02:40

%S 1,1,3,2,3,7,3,9,5,15,5,15,26,3,31,8,30,43,63,-15,63,13,54,104,87,144,

%T -81,127,21,99,203,273,115,333,-275,255,34,177,416,549,609,-9,806,

%U -789,511,55,315,811,1263,1146,1260,-725,2043,-2071,1023,89,555,1573

%N 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 - 3*x - 2*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 A. Higuita, and Antara Mikherjee, <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) 1-28.

%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) = 1 + 3*x, u = p(2,x), and v = 1 - 3*x - 2*x^2.

%F p(n,x) = k*(b^n - c^n), where k = -1/sqrt(5 - 6*x + x^2), b = (1/2)*(3*x + 1 - 1/k), c = (1/2)*(2*x + 1 + 1/k).

%e First eight rows:

%e 1

%e 1 3

%e 2 3 7

%e 3 9 5 15

%e 5 15 26 3 31

%e 8 30 43 63 -15 63

%e 13 54 104 87 144 -81 127

%e 21 99 203 273 115 333 -275 255

%e Row 4 represents the polynomial p(4,x) = 3 + 9*x + 5*x^2 + 15*x^3, so (T(4,k)) = (3,9,5,15), k=0..3.

%t p[1, x_] := 1; p[2, x_] := 1 + 3 x; u[x_] := p[2, x]; v[x_] := 1 - 3x - 2x^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. A000045 (column 1); A000225, (p(n,n-1)); A001787 (row sums), (p(n,1)); A002605 (alternating row sums), (p(n,-1)); A004254, (p(n,-2)); A057084, (p(n,-3)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299, A367300, A367301, A368150, A368151, A368152, A368153, A368154, A368156.

%K sign,tabl

%O 1,3

%A _Clark Kimberling_, Jan 20 2024