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

1,3

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.

Links

Table of n, a(n) for n=1..45.

Example

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).

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 *)

Crossrefs

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

Sequence in context: A187922 A032456 A228442 * A164989 A286473 A358432

Adjacent sequences: A328641 A328642 A328643 * A328645 A328646 A328647

Keyword

nonn,tabl

Author

Clark Kimberling, Nov 03 2019

Status

approved