A327317 Triangular array read by rows: row n shows the coefficients of this polynomial of degree n: p(x,n) = 2^(n-1) ((x+r)^n - (x+s)^n)/(r - s), where r = 2 and s = 1/2.

1, 5, 4, 21, 30, 12, 85, 168, 120, 32, 341, 850, 840, 400, 80, 1365, 4092, 5100, 3360, 1200, 192, 5461, 19110, 28644, 23800, 11760, 3360, 448, 21845, 87376, 152880, 152768, 95200, 37632, 8960, 1024, 87381, 393210, 786384, 917280, 687456, 342720, 112896

Offset

1,2

Comments

p(x,n) is a strong divisibility sequence of polynomials. That is, gcd(p(x,h),p(x,k)) = p(x,gcd(h,k)). 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..43.

Example

First six rows:
     1;
     5,    4;
    21,   30,   12;
    85,  168,  120,   32;
   341,  850,  840,  400,   80;
  1365, 4092, 5100, 3360, 1200, 192;
The first six polynomials, not factored:
1, 5 + 4 x, 21 + 30 x + 12 x^2, 85 + 168 x + 120 x^2 + 32 x^3, 341 + 850 x + 840 x^2 + 400 x^3 + 80 x^4, 1365 + 4092 x + 5100 x^2 + 3360 x^3 + 1200 x^4 + 192 x^5.
The first six polynomials, factored:
1, 5 + 4 x, 3 (7 + 10 x + 4 x^2), (5 + 4 x) (17 + 20 x + 8 x^2), 341 + 850 x + 840 x^2 + 400 x^3 + 80 x^4, 3 (5 + 4 x) (7 + 10 x + 4 x^2) (13 + 10 x + 4 x^2).

Mathematica

r = 2; s = 1/2; f[x_, n_] := 2^(n - 1) ((x + r)^n - (x + s)^n)/(r - s);
Column[Table[Expand[f[x, n]], {n, 1, 5}]]
c[x_, n_] := CoefficientList[Expand[f[x, n]], x]
TableForm[Table[c[x, n], {n, 1, 10}]] (* A327317 array *)
Flatten[Table[c[x, n], {n, 1, 12}]]   (* A327317 sequence *)

Crossrefs

Cf. A327316, A002450 (x=0), A016137 (x=1), A001045 (x = -1), A016162 (x = 2), A016181 (x = 3), A016127) (x = -3), A016157 (x = 1/2).

Sequence in context: A272506 A070374 A057423 * A215139 A320799 A240987

Adjacent sequences: A327314 A327315 A327316 * A327318 A327319 A327320

Keyword

nonn,tabl

Author

Clark Kimberling, Nov 03 2019

Status

approved