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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. 5
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 (list; table; graph; refs; listen; history; text; internal format)
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: A041583 A143915 A309542 * A160820 A158357 A072953

Adjacent sequences:  A327319 A327320 A327321 * A327323 A327324 A327325

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Nov 08 2019

STATUS

approved

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Last modified January 28 10:02 EST 2020. Contains 331319 sequences. (Running on oeis4.)