A327321 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, 3, 7, 18, 27, 5, 21, 27, 27, 61, 300, 630, 540, 405, 91, 549, 1350, 1890, 1215, 729, 547, 3822, 11529, 18900, 19845, 10206, 5103, 205, 1641, 5733, 11529, 14175, 11907, 5103, 2187, 4921, 44280, 177228, 412776, 622566, 612360, 428652, 157464, 59049, 7381

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). 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..46.

Example

p(x,3) = (1/k)((4 (7 + 18 x + 27 x^2))/(3 sqrt(3))), where k = 4/(3 sqrt(3)).
First six rows:
   1;
   1,   3;
   7,  18,   27;
   5,  21,   27,   27;
  61, 300,  630,  540,  405;
  91, 549, 1350, 1890, 1215, 729;
The first six polynomials, not factored:
1, 1 + 3 x, 7 + 18 x + 27 x^2, 5 + 21 x + 27 x^2 + 27 x^3, 61 + 300 x + 630 x^2 + 540 x^3 + 405 x^4, 91 + 549 x + 1350 x^2 + 1890 x^3 + 1215 x^4 + 729 x^5.
The first six polynomials, factored:
1, 1 + 3 x, 7 + 18 x + 27 x^2, (1 + 3 x) (5 + 6 x + 9 x^2), 61 + 300 x + 630 x^2 + 540 x^3 + 405 x^4, (1 + 3 x) (13 + 6 x + 9 x^2) (7 + 18 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]; 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}]]  (* A327321 *)
(* Peter J. C. Moses, Nov 01 2019 *)

Crossrefs

Cf. A327320, A329008, A329000, A031364.

Sequence in context: A102329 A331713 A348391 * A069143 A097007 A308445

Adjacent sequences: A327318 A327319 A327320 * A327322 A327323 A327324

Keyword

nonn,tabl

Author

Clark Kimberling, Nov 08 2019

Status

approved