A134885 Triangular sequence from polynomials that gives roots near 137.

1, 137, -1, -135, -137, 1, 134, 0, 137, -1, -133, 0, 0, -137, 1, 132, 0, 0, 0, 137, -1, -131, 0, 0, 0, 0, -137, 1, 130, 0, 0, 0, 0, 0, 137, -1, -129, 0, 0, 0, 0, 0, 0, -137, 1, 128, 0, 0, 0, 0, 0, 0, 0, 137, -1, -127, 0, 0, 0, 0, 0, 0, 0, 0, -137, 1

Offset

1,2

Links

Table of n, a(n) for n=1..66.

Formula

p(x,0) = 1. p(x,1) = 137-x. p(x,n) = (-1)^(n-1)*(137-n) + (-1)^(n-1)*137*x^(n-1) - (-1)^(n-1)*x^n for n > 1.

T(n,k) = [x^k] p(x,n).

Example

p(x,134) = -3 - 137*x^133 + x^134.
Triangular sequence:
  {1},
  {137, -1},
  {-135, -137, 1},
  {134, 0, 137, -1},
  {-133, 0, 0, -137, 1},
  {132, 0, 0, 0, 137, -1},
  {-131, 0, 0, 0, 0, -137, 1},
  {130, 0, 0, 0, 0, 0, 137, -1},
  ...

Mathematica

p[x_, n_] = (-1)^(n - 1)*(137 - n) + (-1)^(n - 1)*137*x^(n - 1) - (-1)^( n - 1)*x^n
a = Join[{1, 137 - x}, Table[p[x, n], {n, 2, 10}]]
c = Table[CoefficientList[a[[n]], x], {n, 1, Length[a]}]
Flatten[c]

Prog

(PARI) p(x, n) = if(n==0, 1, if(n==1, 137-x, (-1)^(n-1)*(137-n) + (-1)^(n-1)*137*x^(n-1) - (-1)^(n-1)*x^n))
T(n, k) = polcoef(p(x, n), k) \\ Jason Yuen, Jan 28 2025

Crossrefs

Sequence in context: A233254 A001330 A091510 * A259680 A082726 A189998

Adjacent sequences: A134882 A134883 A134884 * A134886 A134887 A134888

Keyword

uned,tabl,sign,less

Author

Roger L. Bagula, Jan 29 2008

Extensions

More terms from Jason Yuen, Jan 28 2025

Status

approved