cp's OEIS Frontend

A368155 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.

%I A368155 #11 Jan 22 2024 00:02:40
%S A368155 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 A368155 -81,127,21,99,203,273,115,333,-275,255,34,177,416,549,609,-9,806,
%U A368155 -789,511,55,315,811,1263,1146,1260,-725,2043,-2071,1023,89,555,1573
%N A368155 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 A368155 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 A368155 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 A368155 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 A368155 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 A368155 First eight rows:
%e A368155    1
%e A368155    1    3
%e A368155    2    3     7
%e A368155    3    9     5    15
%e A368155    5   15    26     3    31
%e A368155    8   30    43    63   -15    63
%e A368155   13   54   104    87   144   -81    127
%e A368155   21   99   203   273   115   333   -275   255
%e A368155 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 A368155 p[1, x_] := 1; p[2, x_] := 1 + 3 x; u[x_] := p[2, x]; v[x_] := 1 - 3x - 2x^2;
%t A368155 p[n_, x_] := Expand[u[x]*p[n - 1, x] + v[x]*p[n - 2, x]]
%t A368155 Grid[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
%t A368155 Flatten[Table[CoefficientList[p[n, x], x], {n, 1, 10}]]
%Y A368155 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 A368155 sign,tabl
%O A368155 1,3
%A A368155 _Clark Kimberling_, Jan 20 2024