|
| |
|
|
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
|
|
|
|
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 *)
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
