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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
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.
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
Sequence in context: A143915 A355597 A309542 * A160820 A158357 A072953
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Nov 08 2019
STATUS
approved