A327318 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 = 1 and s = 1/2.

1, 3, 4, 7, 18, 12, 15, 56, 72, 32, 31, 150, 280, 240, 80, 63, 372, 900, 1120, 720, 192, 127, 882, 2604, 4200, 3920, 2016, 448, 255, 2032, 7056, 13888, 16800, 12544, 5376, 1024, 511, 4590, 18288, 42336, 62496, 60480, 37632, 13824, 2304, 1023, 10220, 45900

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..48.

Example

First six rows:
   1;
   3,   4;
   7,  18,  12;
  15,  56,  72,   32;
  31, 150, 280,  240,  80;
  63, 372, 900, 1120, 720, 192;
The first six polynomials, not factored:
1, 3 + 4 x, 7 + 18 x + 12 x^2, 15 + 56 x + 72 x^2 + 32 x^3, 31 + 150 x + 280 x^2 + 240 x^3 + 80 x^4, 63 + 372 x + 900 x^2 + 1120 x^3 + 720 x^4 + 192 x^5.
The first six polynomials, factored:
1, 3 + 4 x, 7 + 18 x + 12 x^2, (3 + 4 x) (5 + 12 x + 8 x^2), 31 + 150 x + 280 x^2 + 240 x^3 + 80 x^4, (3 + 4 x) (3 + 6 x + 4 x^2) (7 + 18 x + 12 x^2).

Mathematica

r = 1; 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}]] (* A327318 array *)
Flatten[Table[c[x, n], {n, 1, 12}]]   (* A327318 sequence *)

Crossrefs

Cf. A327316, A327317, A000225 (x = 0), A005061 (x = 1), A081199 (x = 1/2).

Sequence in context: A078825 A361659 A331115 * A344783 A093611 A042375

Adjacent sequences: A327315 A327316 A327317 * A327319 A327320 A327321

Keyword

nonn,tabl

Author

Clark Kimberling, Nov 08 2019

Status

approved