A327322 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, 2, 5, 7, 20, 25, 26, 105, 150, 125, 521, 2600, 5250, 5000, 3125, 434, 2605, 6500, 8750, 6250, 3125, 13021, 91140, 273525, 455000, 459375, 262500, 109375, 8138, 65105, 227850, 455875, 568750, 459375, 218750, 78125, 36169, 325520, 1302100, 3038000, 4558750

Offset

1,2

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(5). 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..41.

Example

p(x,3) = (1/k)((18 (7 + 20 x + 25 x^2))/(5 sqrt(5))), where k = 18/(5 sqrt(5)).
First six rows:
    1;
    2,    5;
    7,   20,   25;
   26,  105,  150,  125;
  521, 2600, 5250, 5000, 3125;
  434, 2605, 6500, 8750, 6250, 3125;
The first six polynomials, not factored:
1, 2 + 5 x, 7 + 20 x + 25 x^2, 26 + 105 x + 150 x^2 + 125 x^3, 521 + 2600 x + 5250 x^2 + 5000 x^3 + 3125 x^4, 434 + 2605 x + 6500 x^2 + 8750 x^3 + 6250 x^4 + 3125 x^5.
The first six polynomials, factored:
1, 2 + 5 x, 7 + 20 x + 25 x^2, (2 + 5 x) (13 + 20 x + 25 x^2), 521 + 2600 x + 5250 x^2 + 5000 x^3 + 3125 x^4, (2 + 5 x) (7 + 20 x + 25 x^2) (31 + 20 x + 25 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[5]; 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}]]  (* A327322 *)
(* Peter J. C. Moses, Nov 01 2019 *)

Crossrefs

Cf. A327320, A327321.

Sequence in context: A143915 A355597 A309542 * A160820 A158357 A072953

Adjacent sequences: A327319 A327320 A327321 * A327323 A327324 A327325

Keyword

nonn,tabl

Author

Clark Kimberling, Nov 08 2019

Status

approved