A124032 Triangle read by rows: T(n,k) is the coefficient of x^k in the polynomial p(n,x) defined by p(0,x)=1, p(1,x)=-1-x, p(n,x)=((-1)^(n-1)-x)*p(n-1,x)-p(n-2,x) for n>=2 (0<=k<=n).

1, -1, -1, 0, 2, 1, 1, 3, -1, -1, -1, -6, -3, 2, 1, -2, -8, 4, 6, -1, -1, 3, 16, 7, -12, -6, 2, 1, 5, 21, -13, -25, 7, 9, -1, -1, -8, -42, -15, 50, 24, -18, -9, 2, 1, -13, -55, 40, 90, -33, -51, 10, 12, -1, -1, 21, 110, 30, -180, -81, 102, 50, -24, -12, 2, 1

Offset

0,5

Links

Table of n, a(n) for n=0..65.

Joanne Dombrowski, Tridiagonal matrix representations of cyclic selfadjoint operators, Pacific J. Math. 114, no. 2 (1984), 325-334.

Example

Triangle begins:
  {1},
  {-1, -1},
  {0, 2, 1},
  {1, 3, -1, -1},
  {-1, -6, -3, 2, 1},
  {-2, -8, 4, 6, -1, -1},
  {3, 16, 7, -12, -6, 2, 1},
  {5, 21, -13, -25, 7, 9, -1, -1},
  {-8, -42, -15, 50, 24, -18, -9, 2, 1},
  {-13, -55, 40, 90, -33, -51, 10, 12, -1, -1},
  {21, 110, 30, -180, -81, 102, 50, -24, -12, 2, 1}

Maple

p[0]:=1: p[1]:=-1-x: for n from 2 to 12 do p[n]:=sort(expand(((-1)^(n-1)-x)*p[n-1]-p[n-2])) od: T:=(n, k)->coeff(p[n], x, k): for n from 0 to 12 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form

Mathematica

b[k_] = (-1)^k; a[k_] = -1; p[0, x] = 1; p[1, x] = (x - b[1])/a[1]; p[k_, x_] :=p[k, x] = ((x - b[k - 1])*p[k - 1, x] - a[k - 2] *p[k - 2, x])/a[k - 1]; w = Table[CoefficientList[p[n, x], x], {n, 0, 10}]; Flatten[w]

Crossrefs

Sequence in context: A195825 A098824 A181651 * A254046 A137457 A275699

Adjacent sequences: A124029 A124030 A124031 * A124033 A124034 A124035

Keyword

sign,tabl

Author

Roger L. Bagula and Gary W. Adamson, Nov 01 2006

Extensions

Edited by N. J. A. Sloane, Dec 02 2006

Status

approved