A328644 Triangular array read by rows: row n shows the coefficients of the polynomial p(x,n) constructed as in Comments; these polynomials form a strong divisibility sequence.

1, 1, 6, 7, 9, 27, 13, 84, 54, 108, 11, 39, 126, 54, 81, 133, 990, 1755, 3780, 1215, 1458, 463, 2793, 10395, 12285, 19845, 5103, 5103, 1261, 11112, 33516, 83160, 73710, 95256, 20412, 17496, 4039, 34047, 150012, 301644, 561330, 398034, 428652, 78732, 59049

Offset: 1

Author

Clark Kimberling, Nov 03 2019

Keywords

• nonn
• tabl

Comments

Suppose q is a rational number such that the number r = sqrt(q) is irrational. The function (r x + r)^n - (r x - 1/r)^n of x can be represented as k*p(x,n), where k is a constant and p(x,n) is a product of nonconstant polynomials having GCD = 1; the sequence p(x,n) is a strong divisibility sequence of polynomials; i.e., gcd(p(x,h),p(x,k)) = p(x,gcd(h,k)). For A327320, r = sqrt(3/2). If x is an integer, then p(x,n) is a strong divisibility sequence of integers.

Examples

			We have p(x,3) = (1/k)((5 (7 + 9 x + 27 x^2))/(6 sqrt(6))), where k = 5/(6 sqrt(6)).
First six rows:
  1;
  1, 6;
  7, 9, 27;
  13, 84, 54, 108;
  11, 39, 126, 54, 81;
  133, 990, 1755, 3780, 1215, 1458;
The first six polynomials, not factored:
1, 1 + 6 x, 7 + 9 x + 27 x^2, 13 + 84 x + 54 x^2 + 108 x^3, 11 + 39 x + 126 x^2 + 54 x^3 + 81 x^4, 133 + 990 x + 1755 x^2 + 3780 x^3 + 1215 x^4 + 1458 x^5.
The first six polynomials, factored:
1, 1 + 6 x, 7 + 9 x + 27 x^2, (1 + 6 x) (13 + 6 x + 18 x^2), 11 + 39 x + 126 x^2 + 54 x^3 + 81 x^4, (1 + 6 x) (19 + 3 x + 9 x^2) (7 + 9 x + 27 x^2).
		

Crossrefs

Cf. A327320, A327321, A329017, A329018, A329019.

Programs

Mathematica

c[poly_] := If[Head[poly] === Times, Times @@ DeleteCases[(#1 (Boole[
MemberQ[#1, x] || MemberQ[#1, y] || MemberQ[#1, z]] &) /@
Variables /@ #1 &)[List @@ poly], 0], poly];
r = Sqrt[3/2]; f[x_, n_] := c[Factor[Expand[(r x + r)^n - (r x - 1/r)^n]]];
Table[f[x, n], {n, 1, 6}]
Flatten[Table[CoefficientList[f[x, n], x], {n, 1, 12}]]  (* A328644 *)
(* Peter J. C. Moses, Nov 01 2019 *)