A327323 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, 5, 12, 31, 90, 108, 185, 744, 1080, 864, 1111, 5550, 11160, 10800, 6480, 6665, 39996, 99900, 133920, 97200, 46656, 5713, 39990, 119988, 199800, 200880, 116640, 46656, 239945, 1919568, 6718320, 13438656, 16783200, 13499136, 6531840, 2239488, 1439671

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(6). 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..37.

Example

p(x,3) = (1/k)((7 (31 + 90 x + 108 x^2))/(6 sqrt(6))), where k = 7/(6 sqrt(6)).
First six rows:
     1;
     5,    12;
    31,    90,    108;
   185,   744,   1080,    864;
  1111,  5550,  11160,  10800,   6480;
  6665, 39996,  99900, 133920,  97200,  46656;
  5713, 39990, 119988, 199800, 200880, 116640, 46656;
The first six polynomials, not factored:
1, 5 + 12 x, 31 + 90 x + 108 x^2, 185 + 744 x + 1080 x^2 + 864 x^3, 1111 + 5550 x + 11160 x^2 + 10800 x^3 + 6480 x^4, 6665 + 39996 x + 99900 x^2 + 133920 x^3 + 97200 x^4 + 46656 x^5.
The first six polynomials, factored:
1, 5 + 12 x, 31 + 90 x + 108 x^2, (5 + 12 x) (37 + 60 x + 72 x^2), 1111 + 5550 x +  11160 x^2 + 10800 x^3 + 6480 x^4, (5 + 12 x) (43 + 30 x + 36 x^2) (31 + 90 x + 108 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[6]; 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}]]  (* A327323 *)
(* Peter J. C. Moses, Nov 01 2019 *)

Crossrefs

Cf. A327320, A327321, A327322, A329014, A329015, A329016.

Sequence in context: A090974 A038606 A361224 * A192303 A301785 A066280

Adjacent sequences: A327320 A327321 A327322 * A327324 A327325 A327326

Keyword

nonn,tabl

Author

Clark Kimberling, Nov 09 2019

Status

approved