1,3

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.

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).

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 *)

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

Sequence in context: A187922 A032456 A228442 * A164989 A286473 A358432

Adjacent sequences: A328641 A328642 A328643 * A328645 A328646 A328647

nonn,tabl

Clark Kimberling, Nov 03 2019

approved