A328644 - OEIS

A328644

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, 6, 7, 9, 27, 13, 84, 54, 108, 11, 39, 126, 54, 81, 133, 990, 1755, 3780, 1215, 1458, 463, 2793, 10395, 12285, 19845, 5103, 5103, 1261, 11112, 33516, 83160, 73710, 95256, 20412, 17496, 4039, 34047, 150012, 301644, 561330, 398034, 428652, 78732, 59049

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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(3/2). 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..45.

EXAMPLE

We have p(x,3) = (1/k)((5 (7 + 9 x + 27 x^2))/(6 sqrt(6))), where k = 5/(6 sqrt(6)).

First six rows:

1, 6;

7, 9, 27;

13, 84, 54, 108;

11, 39, 126, 54, 81;

133, 990, 1755, 3780, 1215, 1458;

The first six polynomials, not factored:

1, 1 + 6 x, 7 + 9 x + 27 x^2, 13 + 84 x + 54 x^2 + 108 x^3, 11 + 39 x + 126 x^2 + 54 x^3 + 81 x^4, 133 + 990 x + 1755 x^2 + 3780 x^3 + 1215 x^4 + 1458 x^5.

The first six polynomials, factored:

1, 1 + 6 x, 7 + 9 x + 27 x^2, (1 + 6 x) (13 + 6 x + 18 x^2), 11 + 39 x + 126 x^2 + 54 x^3 + 81 x^4, (1 + 6 x) (19 + 3 x + 9 x^2) (7 + 9 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[3/2]; 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}]] (* A328644 *)

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

CROSSREFS

Cf. A327320, A327321, A329017, A329018, A329019.