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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. 4
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 (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(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: A038357 A090974 A038606 * A192303 A301785 A066280

Adjacent sequences:  A327320 A327321 A327322 * A327324 A327325 A327326

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Nov 09 2019

STATUS

approved

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Last modified August 11 03:41 EDT 2022. Contains 356046 sequences. (Running on oeis4.)