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A327320 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. 11
1, 1, 4, 1, 2, 4, 5, 24, 24, 32, 11, 50, 120, 80, 80, 7, 44, 100, 160, 80, 64, 43, 294, 924, 1400, 1680, 672, 448, 85, 688, 2352, 4928, 5600, 5376, 1792, 1024, 19, 170, 688, 1568, 2464, 2240, 1792, 512, 256, 341, 3420, 15300, 41280, 70560, 88704, 67200 (list; table; graph; refs; listen; history; text; internal format)
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(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..52.

EXAMPLE

p(x,3) = (1/k)(9 (1 + 2 x + 4 x^2))/(2 sqrt(2)), where k = 9/(2 sqrt(2)).

First six rows:

1;

1, 4;

1, 2, 4;

5, 24, 24, 32;

11, 50, 120, 80, 80;

7, 44, 100, 160, 80, 64;

The first six polynomials, not factored:

1, 1 + 4 x, 1 + 2 x + 4 x^2, 5 + 24 x + 24 x^2 + 32 x^3, 11 + 50 x + 120 x^2 + 80 x^3 + 80 x^4, 7 + 44 x + 100 x^2 + 160 x^3 + 80 x^4 + 64 x^5.

The first six polynomials, factored:

1, 1 + 4 x, 1 + 2 x + 4 x^2, (1 + 4 x) (5 + 4 x + 8 x^2), 11 + 50 x + 120 x^2 + 80 x^3 + 80 x^4, (1 + 4 x) (1 + 2 x + 4 x^2) (7 + 2 x + 4 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[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}]] (* A327320 *)

(* Peter J. C. Moses, Nov 01 2019 *)

CROSSREFS

Cf. A327315.

Sequence in context: A307550 A309443 A014571 * A324466 A152523 A082903

Adjacent sequences: A327317 A327318 A327319 * A327321 A327322 A327323

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Nov 08 2019

STATUS

approved

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Last modified January 29 17:05 EST 2023. Contains 359923 sequences. (Running on oeis4.)