A367300 - OEIS

A367300

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

%I #11 Dec 25 2023 13:47:14

%S 1,3,2,10,10,3,33,46,22,4,109,194,131,40,5,360,780,678,296,65,6,1189,

%T 3036,3228,1828,581,98,7,3927,11546,14514,10100,4194,1036,140,8,12970,

%U 43150,62601,51664,26479,8604,1722,192,9,42837,159082,261598,249720

%N Triangular array T(n,k), read by rows: coefficients of strong divisibility sequence of polynomials p(1,x) = 1, p(2,x) = 3 + 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 - 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 + 2*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 + 4*x)), b = (1/2) (2*x + 3 + 1/k), c = (1/2) (2*x + 3 - 1/k).

%e First eight rows:

%e 1 3 2

%e 10 10 3

%e 33 46 22 4

%e 109 194 131 40 5

%e 360 780 678 296 65 6

%e 1189 3036 3228 1828 581 98 7

%e 3927 11546 14514 10100 4194 1036 140 8

%e Row 4 represents the polynomial p(4,x) = 33 + 46*x + 22*x^2 + 4*x^3, so (T(4,k)) = (33,46,22,4), k=0..3.

%t p[1, x_] := 1; p[2, x_] := 3 + 2 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); A000027 (p(n,n-1)); A107839 (row sums, (p(n,1)); A001045 (alternating row sums), (p(n,-1)); A030240 (p(n,2)); A039834 (signed Fibonacci numbers), (p(n,-2)); A016130 (p(n,3)); A225883 (p(n,-3)); A099450 (p(n,-4)); A094440, A367208, A367209, A367210, A367211, A367297, A367298, A367299.

%K nonn,tabf

%O 1,2

%A _Clark Kimberling_, Dec 23 2023