A137454 Irregular triangle read by rows: coefficients of polynomials p(x, n) where p(x,n) = x^2*p(x,n-1) + (-x-1)*p(x,n-2) + p(x,n-3).

1, -1, -1, 1, -1, -1, -1, -1, 1, 2, 2, -1, -2, -1, -1, 1, 0, 1, 5, 4, -1, -3, -1, -1, 1, -3, -5, -2, 3, 9, 6, -1, -4, -1, -1, 1, 2, 1, -10, -16, -6, 6, 14, 8, -1, -5, -1, -1, 1, 3, 9, 14, 4, -23, -34, -12, 10, 20, 10, -1, -6, -1, -1, 1, -5, -8, 10, 38, 45, 10, -44, -60, -20, 15, 27, 12, -1, -7, -1, -1, 1, -1, -11, -38, -42, 23, 101, 105, 20

Offset: 0

Author

Roger L. Bagula and Gary W. Adamson, Apr 18 2008

Keywords

• tabf
• easy
• sign

Examples

			{1},
{-1, -1, 1},
{-1, -1, -1, -1, 1},
{2, 2, -1, -2, -1, -1, 1},
{0, 1, 5, 4, -1, -3, -1, -1, 1},
{-3, -5, -2, 3,9, 6, -1, -4, -1, -1, 1},
{2, 1, -10, -16, -6, 6,14, 8, -1, -5, -1, -1, 1},
{3, 9,14, 4, -23, -34, -12, 10, 20, 10, -1, -6, -1, -1, 1},
{-5, -8, 10, 38, 45, 10, -44, -60, -20, 15,27, 12, -1, -7, -1, -1, 1},
{-1, -11, -38, -42, 23, 101, 105,20, -75, -95, -30, 21, 35, 14, -1, -8, -1, -1,1},
{8, 22, 11, -55, -144, -131, 45, 215, 205, 35, -118, -140, -42, 28, 44, 16, -1, -9, -1, -1, 1}
		

Crossrefs

Cf. A084610.

Programs

Mathematica

p[x, -1] = 0;
p[x, 0] = 1;
p[x, 1] = x^2 - x - 1;
p[x_, n_] := x^2*p[x, n - 1] + (-x - 1)*p[x, n - 2] + p[x, n - 3];
Table[Expand[p[x, n]], {n, 0, 10}];
a = Table[CoefficientList[p[x, n], x], {n, 0, 10}];
Flatten[a1]

Formula

p(x,-1)=0; p(x,0)=1; p(x,1) = x^2 - x - 1; p(x,n) = x^2*p(x,n-1) + (-x-1)*p(x,n-2) + p(x,n-3).

Extensions

Heavily edited and corrected by Joerg Arndt, Apr 30 2018