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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
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.
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
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Nov 08 2019
STATUS
approved